2406/2406.00575.md

Title: Existence-Uniqueness Theory and Small-Data Decay for a Reaction-Diffusion Model of Wildfire Spread

URL Source: https://arxiv.org/html/2406.00575

Published Time: Fri, 07 Jun 2024 01:04:50 GMT

Markdown Content: Existence-Uniqueness Theory and Small-Data Decay for a Reaction-Diffusion Model of Wildfire Spread

1. 1 Introduction 1. 1.1 Notation

2. 2 Results 1. 2.1 Preliminaries 2. 2.2 Mild Solutions and their Basic Properties 3. 2.3 A Priori Estimates and Existence-Uniqueness Theory 4. 2.4 Decay of Small-Data Solutions

3. 3 Acknowledgements

Existence-Uniqueness Theory and Small-Data Decay for a Reaction-Diffusion Model of Wildfire Spread

A. George Morgan 1 1 1 Department of Mathematics, University of Toronto, 40 St. George St., Room 6290, Toronto, Ontario, CA, M5S 2E4. Institutional email: adam.morgan@mail.utoronto.ca

Abstract

I examine some analytical properties of a nonlinear reaction-diffusion system that has been used to model the propagation of a wildfire. I establish global-in-time existence and uniqueness of bounded mild solutions to the Cauchy problem for this system given bounded initial data. In particular, this shows that the model does not allow for thermal blow-up. If the initial temperature and fuel density also satisfy certain integrability conditions, the L 2 superscript 𝐿 2 L^{2}italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT-norms of these global solutions are uniformly bounded in time. Additionally, I use a bootstrap argument to show that small initial temperatures give rise to solutions that decay to zero as time goes to infinity, proving the existence of initial states that do not develop into travelling combustion waves.

Keywords– Wildfire spread models, solid-fuel combustion, reaction-diffusion equations in combustion theory, existence and uniqueness of solutions to reaction-diffusion equations, time decay of solutions to reaction-diffusion equations

MSC classes– 35K57, 80A25

1 Introduction

In this article, I consider a reaction-diffusion model of wildfire propagation in a fuel layer with a very large spatial extent. Fix a spatial dimension d≥1 𝑑 1 d\geq 1 italic_d ≥ 1. Consider a certain amount of solid fuel (trees, shrubberies, and so on) spread throughout ℝ d superscript ℝ 𝑑\mathbb{R}^{d}blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT with dimensionless mass density at spatial location x 𝑥 x italic_x and time t 𝑡 t italic_t given by Y⁢(x,t)𝑌 𝑥 𝑡 Y(x,t)italic_Y ( italic_x , italic_t ). A fire with a dimensionless temperature field T⁢(x,t)𝑇 𝑥 𝑡 T(x,t)italic_T ( italic_x , italic_t ) is moving through space and consuming this fuel: temperature is normalized so that T=0 𝑇 0 T=0 italic_T = 0 is the ambient temperature of the system’s environment. Also, we suppose the reaction rate governing combustion is given by the Arrhenius law

r⁢(T)={e−1 T T>0 0 T≤0.𝑟 𝑇 cases superscript 𝑒 1 𝑇 𝑇 0 0 𝑇 0 r(T)=\begin{cases}e^{-\frac{1}{T}}&\quad T>0\ 0&\quad T\leq 0.\end{cases}italic_r ( italic_T ) = { start_ROW start_CELL italic_e start_POSTSUPERSCRIPT - divide start_ARG 1 end_ARG start_ARG italic_T end_ARG end_POSTSUPERSCRIPT end_CELL start_CELL italic_T > 0 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL italic_T ≤ 0 . end_CELL end_ROW(1.1)

Under these assumptions, Weber et al. [19] were the first to model the evolution of T 𝑇 T italic_T and Y 𝑌 Y italic_Y using the following Cauchy problem: given two parameters λ,β≥0 𝜆 𝛽 0\lambda,\beta\geq 0 italic_λ , italic_β ≥ 0,

{T t=(Δ−λ)⁢T+Y⁢r⁢(T)Y t=−β⁢Y⁢r⁢(T)T|t=0=T 0⁢(x)Y|t=0=Y 0⁢(x).\left{\begin{aligned} T_{t}&=\left(\Delta-\lambda\right)T+Yr\left(T\right)\ Y_{t}&=-\beta Yr\left(T\right)\ T|{t=0}&=T{0}(x)\ Y|{t=0}&=Y{0}(x).\end{aligned}\right.{ start_ROW start_CELL italic_T start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT end_CELL start_CELL = ( roman_Δ - italic_λ ) italic_T + italic_Y italic_r ( italic_T ) end_CELL end_ROW start_ROW start_CELL italic_Y start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT end_CELL start_CELL = - italic_β italic_Y italic_r ( italic_T ) end_CELL end_ROW start_ROW start_CELL italic_T | start_POSTSUBSCRIPT italic_t = 0 end_POSTSUBSCRIPT end_CELL start_CELL = italic_T start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_x ) end_CELL end_ROW start_ROW start_CELL italic_Y | start_POSTSUBSCRIPT italic_t = 0 end_POSTSUBSCRIPT end_CELL start_CELL = italic_Y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_x ) . end_CELL end_ROW(1.2)

We always assume Y 0⁢(x)≥0 subscript 𝑌 0 𝑥 0 Y_{0}(x)\geq 0 italic_Y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_x ) ≥ 0 everywhere. Up to a different choice of reaction rate, these are precisely the equations of first-order solid-fuel combustion at infinite Lewis number, with an added heat loss term [3, §1.3]. Strictly speaking, in [19] and other work Y⁢(x,t)𝑌 𝑥 𝑡 Y(x,t)italic_Y ( italic_x , italic_t ) is actually the mass fraction of fuel. However, I want to allow for an infinite initial amount of fuel, and in this situation the mass fraction does not make sense and it is more natural to regard Y⁢(x,t)𝑌 𝑥 𝑡 Y(x,t)italic_Y ( italic_x , italic_t ) as a mass density. For an instructive derivation of (1.2) and an explanation of why it is useful for data assimilation, see Mandel et al. [11]. The discussion in Mandel et al. highlights that r⁢(0)=0 𝑟 0 0 r(0)=0 italic_r ( 0 ) = 0 implies no combustion takes place at the ambient temperature T=0 𝑇 0 T=0 italic_T = 0. Consequently, (1.2) admits solutions that take the form of travelling combustion waves; this had already been confirmed using formal asymptotics and numerics in [19].

In light of the existence of travelling wave solutions, (1.2) has attracted attention from researchers in both mathematical modelling and dynamical systems theory. On the more concrete side, (1.2) has been used to develop computationally efficient reduced-order models of wildfire dynamics [5, 9]. Also, Johnston [8] applied (1.2) in a modelling pipeline where remote sensing data was leveraged to estimate realistic model parameter values. Additionally, variants of the model incorporating the effects of wind [2, 11] and radiative heat transfer [1, 12, 15, 18] have also been considered. When it comes to rigorous results on the behaviour of solutions to (1.2), Billingham [4] proved the existence of travelling waves given suitable initial data and d=1 𝑑 1 d=1 italic_d = 1. Linear stability of these travelling waves was examined numerically by Varas and Vega [17]. Existence and linear stability of travelling waves for related combustion models in d=1 𝑑 1 d=1 italic_d = 1 with finite Lewis number has also been considered, see for example [7, 14]. There is certainly no shortage of other interesting work on travelling waves in wildland fire propagation and other solid-fuel systems, but a complete literature review is beyond the scope of this brief article.

While there is plenty of mathematical work on the properties of travelling wave solutions to (1.2), to my knowledge there are no published results on the existence and uniqueness of global-in-time solutions to (1.2) for generic bounded initial data (see, however, the existence result for a related model in [3, §4.4]). In particular, with the exception of the lumped-parameter analysis in [15], I have not found any estimates that preclude finite-time blow-up of |T⁢(x,t)|𝑇 𝑥 𝑡|T(x,t)|| italic_T ( italic_x , italic_t ) |. Thermal blow-up phenomena indeed arise in certain solid-fuel combustion models [3, §3.2], so understanding whether or not blow-up occurs in solutions to (1.2) is a meaningful question. Also, in [11], the authors use a heuristic lumped-parameter argument to predict that a fire with a sufficiently low temperature eventually burns itself out: in the jargon of combustion theory, this means the model predicts a nonzero “auto-ignition temperature”. I have not, however, found any rigorous proof of the claim that (1.2) has a nonzero auto-ignition temperature.

In this article, I prove three novel results to address the knowledge gaps discussed in the previous paragraph:

1. (i)for any bounded initial state (T 0,Y 0)subscript 𝑇 0 subscript 𝑌 0\left(T_{0},Y_{0}\right)( italic_T start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_Y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) there exists a unique, bounded mild solution to (1.2) (to be defined precisely below) valid on the entire time interval [0,∞)0[0,\infty)[ 0 , ∞ ), provided λ>0 𝜆 0\lambda>0 italic_λ > 0: in particular, thermal blow-up is impossible (corollary 2.7);
2. (ii)further, if Y 0 subscript 𝑌 0 Y_{0}italic_Y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT is also integrable and T 0 subscript 𝑇 0 T_{0}italic_T start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT is also square-integrable, then the L x 2 subscript superscript 𝐿 2 𝑥 L^{2}_{x}italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT norm of T⁢(x,t)𝑇 𝑥 𝑡 T(x,t)italic_T ( italic_x , italic_t ) is bounded uniformly in time (corollary 2.7 as well);
3. (iii)given an initial state T 0 subscript 𝑇 0 T_{0}italic_T start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT that is small in a particular physically reasonable norm, the solution to (1.2) satisfies lim t→0|T⁢(x,t)|=0 subscript→𝑡 0 𝑇 𝑥 𝑡 0\lim_{t\rightarrow 0}|T(x,t)|=0 roman_lim start_POSTSUBSCRIPT italic_t → 0 end_POSTSUBSCRIPT | italic_T ( italic_x , italic_t ) | = 0 uniformly in x 𝑥 x italic_x for every λ≥0 𝜆 0\lambda\geq 0 italic_λ ≥ 0: morally, this confirms that (1.2) indeed has a nonzero auto-ignition temperature, and that not every initial state gives rise to a travelling wave solution (theorem 2.10);

I hope these results prove useful to further investigations into the nonlinear stability of travelling waves and to the development of rigorous error estimates for numerical discretizations of (1.2).

1.1 Notation

For definitions of any of the objects appearing below, see [6].

• •d 𝑑 d italic_d always denotes a positive integer greater than or equal to 1 1 1 1.
• •ℕ 0={0,1,2,3,…}subscript ℕ 0 0 1 2 3…\mathbb{N}_{0}=\left{0,1,2,3,...\right}blackboard_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = { 0 , 1 , 2 , 3 , … }.
• •C 𝐶 C italic_C always denotes some nonnegative constant that may change from line to line. If α 𝛼\alpha italic_α is some real parameter, then C⁢(α)𝐶 𝛼 C(\alpha)italic_C ( italic_α ) denotes a constant depending on α 𝛼\alpha italic_α. If a,b∈ℝ 𝑎 𝑏 ℝ a,b\in\mathbb{R}italic_a , italic_b ∈ blackboard_R, we say that a≲b less-than-or-similar-to 𝑎 𝑏 a\lesssim b italic_a ≲ italic_b if there exists C 𝐶 C italic_C such that a≤b⁢C 𝑎 𝑏 𝐶 a\leq bC italic_a ≤ italic_b italic_C. If a,b∈ℝ 𝑎 𝑏 ℝ a,b\in\mathbb{R}italic_a , italic_b ∈ blackboard_R, we say that a≃b similar-to-or-equals 𝑎 𝑏 a\simeq b italic_a ≃ italic_b if a≲b less-than-or-similar-to 𝑎 𝑏 a\lesssim b italic_a ≲ italic_b and b≲a less-than-or-similar-to 𝑏 𝑎 b\lesssim a italic_b ≲ italic_a.
• •For k∈ℕ 0 𝑘 subscript ℕ 0 k\in\mathbb{N}{0}italic_k ∈ blackboard_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and a nice function φ:ℝ d→ℝ:𝜑→superscript ℝ 𝑑 ℝ\varphi\colon\mathbb{R}^{d}\rightarrow\mathbb{R}italic_φ : blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT → blackboard_R, D x k⁢φ subscript superscript 𝐷 𝑘 𝑥 𝜑 D^{k}{x}\varphi italic_D start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT italic_φ denotes the set of all order k 𝑘 k italic_k partial derivatives of φ 𝜑\varphi italic_φ.
• •For p∈[1,∞]𝑝 1 p\in\left[1,\infty\right]italic_p ∈ [ 1 , ∞ ], L x p⁢(ℝ d)subscript superscript 𝐿 𝑝 𝑥 superscript ℝ 𝑑 L^{p}_{x}\left(\mathbb{R}^{d}\right)italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ) denotes the usual Lebesgue space of real-valued functions of x∈ℝ d 𝑥 superscript ℝ 𝑑 x\in\mathbb{R}^{d}italic_x ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT.
• •For k∈ℕ 0 𝑘 subscript ℕ 0 k\in\mathbb{N}{0}italic_k ∈ blackboard_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, H x k⁢(ℝ d)subscript superscript 𝐻 𝑘 𝑥 superscript ℝ 𝑑 H^{k}{x}\left(\mathbb{R}^{d}\right)italic_H start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ) denotes the L x 2 subscript superscript 𝐿 2 𝑥 L^{2}{x}italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT-based inhomogeneous order k 𝑘 k italic_k Sobolev space of real-valued functions on ℝ d superscript ℝ 𝑑\mathbb{R}^{d}blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT, and H˙x k⁢(ℝ d)subscript superscript˙𝐻 𝑘 𝑥 superscript ℝ 𝑑\dot{H}^{k}{x}\left(\mathbb{R}^{d}\right)over˙ start_ARG italic_H end_ARG start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ) denotes the L x 2 subscript superscript 𝐿 2 𝑥 L^{2}_{x}italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT-based homogeneous order k 𝑘 k italic_k Sobolev space of real-valued functions on ℝ d superscript ℝ 𝑑\mathbb{R}^{d}blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT.
• •Given a Banach space A 𝐴 A italic_A and a time interval [0,t∗]0 subscript 𝑡[0,t_{}][ 0 , italic_t start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ], we let C t 0⁢A subscript superscript 𝐶 0 𝑡 𝐴 C^{0}{t}A italic_C start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT italic_A denote the space of continuous curves [0,t∗]→A→0 subscript 𝑡 𝐴[0,t{}]\rightarrow A[ 0 , italic_t start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ] → italic_A. When the value of t∗subscript 𝑡 t_{*}italic_t start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT has a substantial impact on the topic at hand, this value will be made clear.

2 Results

2.1 Preliminaries

First, we recall some decay and smoothing estimates for the linear heat equation.

Lemma 2.1.

1. (i)Fix any p∈[1,∞]𝑝 1 p\in[1,\infty]italic_p ∈ [ 1 , ∞ ]. For any φ∈L x p 𝜑 subscript superscript 𝐿 𝑝 𝑥\varphi\in L^{p}_{x}italic_φ ∈ italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT,

‖e t⁢Δ⁢φ‖L x p≤‖φ‖L x p⁢∀t>0.subscript norm superscript 𝑒 𝑡 Δ 𝜑 subscript superscript 𝐿 𝑝 𝑥 subscript norm 𝜑 subscript superscript 𝐿 𝑝 𝑥 for-all 𝑡 0\left|e^{t\Delta}\varphi\right|{L^{p}{x}}\leq\left|\varphi\right|{L^{p}% {x}}\quad\forall\ t>0.∥ italic_e start_POSTSUPERSCRIPT italic_t roman_Δ end_POSTSUPERSCRIPT italic_φ ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT ≤ ∥ italic_φ ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∀ italic_t > 0 .(2.1) 2. (ii)For any p,q,ℓ∈[1,∞]𝑝 𝑞 ℓ 1 p,q,\ell\in\left[1,\infty\right]italic_p , italic_q , roman_ℓ ∈ [ 1 , ∞ ] with p−1=q−1−ℓ−1 superscript 𝑝 1 superscript 𝑞 1 superscript ℓ 1 p^{-1}=q^{-1}-\ell^{-1}italic_p start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT = italic_q start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT - roman_ℓ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT, any φ∈L x q 𝜑 subscript superscript 𝐿 𝑞 𝑥\varphi\in L^{q}{x}italic_φ ∈ italic_L start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT, and any k∈ℕ 0 𝑘 subscript ℕ 0 k\in\mathbb{N}{0}italic_k ∈ blackboard_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, we have

‖D x k⁢e t⁢Δ⁢φ‖L x p≲t−1 2⁢(d ℓ+k)⁢‖φ‖L x q⁢∀t>0.less-than-or-similar-to subscript norm subscript superscript 𝐷 𝑘 𝑥 superscript 𝑒 𝑡 Δ 𝜑 subscript superscript 𝐿 𝑝 𝑥 superscript 𝑡 1 2 𝑑 ℓ 𝑘 subscript norm 𝜑 subscript superscript 𝐿 𝑞 𝑥 for-all 𝑡 0\left|D^{k}{x}e^{t\Delta}\varphi\right|{L^{p}{x}}\lesssim t^{-\frac{1}{2}% \left(\frac{d}{\ell}+k\right)}\left|\varphi\right|{L^{q}_{x}}\quad\forall\ % t>0.∥ italic_D start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT italic_e start_POSTSUPERSCRIPT italic_t roman_Δ end_POSTSUPERSCRIPT italic_φ ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT ≲ italic_t start_POSTSUPERSCRIPT - divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( divide start_ARG italic_d end_ARG start_ARG roman_ℓ end_ARG + italic_k ) end_POSTSUPERSCRIPT ∥ italic_φ ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∀ italic_t > 0 .(2.2)

Proof.

All these bounds follow from the time-decay properties of the heat kernel on ℝ d superscript ℝ 𝑑\mathbb{R}^{d}blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT; for example, see [10, pp. 41-42] for an outline of how to prove (2.2). ∎

Additionally, we need a result about the function r⁢(T)𝑟 𝑇 r(T)italic_r ( italic_T ) defined in (1.1).

Lemma 2.2.

For any p∈ℕ 0 𝑝 subscript ℕ 0 p\in\mathbb{N}_{0}italic_p ∈ blackboard_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, there exists a constant C⁢(p)𝐶 𝑝 C(p)italic_C ( italic_p ) such that

r⁢(T)≤C⁢(p)⁢T p⁢∀T∈[0,1].𝑟 𝑇 𝐶 𝑝 superscript 𝑇 𝑝 for-all 𝑇 0 1 r(T)\leq C(p)T^{p}\quad\forall\ T\in[0,1].italic_r ( italic_T ) ≤ italic_C ( italic_p ) italic_T start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ∀ italic_T ∈ [ 0 , 1 ] .

Proof.

This is obvious since r⁢(T)𝑟 𝑇 r(T)italic_r ( italic_T ) vanishes to all orders at T=0 𝑇 0 T=0 italic_T = 0: it is the canonical example of a smooth, non-analytic function. ∎

2.2 Mild Solutions and their Basic Properties

We may use Duhamel’s principle to re-formulate (1.2) in a way that lets us introduce solutions with limited regularity:

Definition 2.3.

Let A,A 0 𝐴 subscript 𝐴 0 A,A_{0}italic_A , italic_A start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and B 𝐵 B italic_B be Banach subspaces of L x∞subscript superscript 𝐿 𝑥 L^{\infty}{x}italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT, with A⊆A 0 𝐴 subscript 𝐴 0 A\subseteq A{0}italic_A ⊆ italic_A start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT. Suppose we are given an initial datum (T 0⁢(x),Y 0⁢(x))∈A 0×B subscript 𝑇 0 𝑥 subscript 𝑌 0 𝑥 subscript 𝐴 0 𝐵\left(T_{0}(x),Y_{0}(x)\right)\in A_{0}\times B( italic_T start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_x ) , italic_Y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_x ) ) ∈ italic_A start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT × italic_B. We say that (T⁢(x,t),Y⁢(x,t))∈C t 0⁢A×C t 0⁢B 𝑇 𝑥 𝑡 𝑌 𝑥 𝑡 subscript superscript 𝐶 0 𝑡 𝐴 subscript superscript 𝐶 0 𝑡 𝐵\left(T(x,t),Y(x,t)\right)\in C^{0}{t}A\times C^{0}{t}B( italic_T ( italic_x , italic_t ) , italic_Y ( italic_x , italic_t ) ) ∈ italic_C start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT italic_A × italic_C start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT italic_B is a mild solution of (1.2) if this pair satisfies

T⁢(x,t)𝑇 𝑥 𝑡\displaystyle T(x,t)italic_T ( italic_x , italic_t )=e−λ⁢t⁢e t⁢Δ⁢T 0+∫0 t e−λ⁢(t−τ)⁢e(t−τ)⁢Δ⁢Y⁢(x,τ)⁢r⁢(T⁢(x,τ))⁢d τ absent superscript 𝑒 𝜆 𝑡 superscript 𝑒 𝑡 Δ subscript 𝑇 0 superscript subscript 0 𝑡 superscript 𝑒 𝜆 𝑡 𝜏 superscript 𝑒 𝑡 𝜏 Δ 𝑌 𝑥 𝜏 𝑟 𝑇 𝑥 𝜏 differential-d 𝜏\displaystyle=e^{-\lambda t}e^{t\Delta}T_{0}+\int_{0}^{t}e^{-\lambda\left(t-% \tau\right)}e^{\left(t-\tau\right)\Delta}Y(x,\tau)r\left(T(x,\tau)\right)\ % \mathrm{d}\tau= italic_e start_POSTSUPERSCRIPT - italic_λ italic_t end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT italic_t roman_Δ end_POSTSUPERSCRIPT italic_T start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT - italic_λ ( italic_t - italic_τ ) end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT ( italic_t - italic_τ ) roman_Δ end_POSTSUPERSCRIPT italic_Y ( italic_x , italic_τ ) italic_r ( italic_T ( italic_x , italic_τ ) ) roman_d italic_τ(2.3a) Y⁢(x,t)𝑌 𝑥 𝑡\displaystyle Y(x,t)italic_Y ( italic_x , italic_t )=Y 0⁢(x)⁢exp⁡(−β⁢∫0 t r⁢(T⁢(x,σ))⁢d σ).absent subscript 𝑌 0 𝑥 𝛽 superscript subscript 0 𝑡 𝑟 𝑇 𝑥 𝜎 differential-d 𝜎\displaystyle=Y_{0}(x)\exp\left(-\beta\int_{0}^{t}r\left(T(x,\sigma)\right)\ % \mathrm{d}\sigma\right).= italic_Y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_x ) roman_exp ( - italic_β ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT italic_r ( italic_T ( italic_x , italic_σ ) ) roman_d italic_σ ) .(2.3b)

Below, the time interval on which a mild solution is valid will be specified when it’s vital to know. Immediately, one sees that mild solutions have some important properties.

1. (i)First, if Y 0⁢(x)≥0 subscript 𝑌 0 𝑥 0 Y_{0}(x)\geq 0 italic_Y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_x ) ≥ 0 everywhere then Y⁢(x,t)≥0 𝑌 𝑥 𝑡 0 Y(x,t)\geq 0 italic_Y ( italic_x , italic_t ) ≥ 0 also holds. This makes perfect sense: an initially nonnegative fuel density should never become negative.
2. (ii)Second, owing to the smoothing properties of the heat flow e t⁢Δ superscript 𝑒 𝑡 Δ e^{t\Delta}italic_e start_POSTSUPERSCRIPT italic_t roman_Δ end_POSTSUPERSCRIPT, T⁢(x,t)𝑇 𝑥 𝑡 T(x,t)italic_T ( italic_x , italic_t ) is smooth in x 𝑥 x italic_x for every t>0 𝑡 0 t>0 italic_t > 0. However, Y⁢(x,t)𝑌 𝑥 𝑡 Y(x,t)italic_Y ( italic_x , italic_t ) is only as smooth as Y 0⁢(x)subscript 𝑌 0 𝑥 Y_{0}(x)italic_Y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_x ).

2.3 A Priori Estimates and Existence-Uniqueness Theory

In this subsection I prove some basic a priori bounds on the Lebesgue and Sobolev norms of mild solutions to (1.2) with λ>0 𝜆 0\lambda>0 italic_λ > 0. These bounds are then used to help construct global-in-time mild solutions of (1.2). We start with the easiest bound possible:

Proposition 2.4.

For any p∈[1,∞]𝑝 1 p\in[1,\infty]italic_p ∈ [ 1 , ∞ ], if (T,Y)𝑇 𝑌(T,Y)( italic_T , italic_Y ) is a mild solution of (1.2), then

sup t>0‖Y⁢(x,t)‖L x p≤‖Y 0⁢(x)‖L x p subscript supremum 𝑡 0 subscript norm 𝑌 𝑥 𝑡 subscript superscript 𝐿 𝑝 𝑥 subscript norm subscript 𝑌 0 𝑥 subscript superscript 𝐿 𝑝 𝑥\sup_{t>0}\left|Y(x,t)\right|{L^{p}{x}}\leq\left|Y_{0}(x)\right|{L^{p}% {x}}roman_sup start_POSTSUBSCRIPT italic_t > 0 end_POSTSUBSCRIPT ∥ italic_Y ( italic_x , italic_t ) ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT ≤ ∥ italic_Y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_x ) ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT(2.4)

provided the right-hand side is finite.

Proof.

Since r⁢(T)𝑟 𝑇 r(T)italic_r ( italic_T ) is nonnegative, Y⁢(x,t)≤Y 0⁢(x)𝑌 𝑥 𝑡 subscript 𝑌 0 𝑥 Y(x,t)\leq Y_{0}(x)italic_Y ( italic_x , italic_t ) ≤ italic_Y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_x ) almost everywhere. ∎

According to Mandel et al. [11, §4.1], Y⁢(x,t)𝑌 𝑥 𝑡 Y(x,t)italic_Y ( italic_x , italic_t ) is expected to settle to a nonzero steady value for t≫1 much-greater-than 𝑡 1 t\gg 1 italic_t ≫ 1. I therefore do not think one can prove a variant of the above result that includes a time-dependent decay factor on the right-hand side.

Proposition 2.5.

Suppose λ>0 𝜆 0\lambda>0 italic_λ > 0. If we are given an initial datum (T 0,Y 0)∈L x∞×L x∞subscript 𝑇 0 subscript 𝑌 0 subscript superscript 𝐿 𝑥 subscript superscript 𝐿 𝑥(T_{0},Y_{0})\in L^{\infty}{x}\times L^{\infty}{x}( italic_T start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_Y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) ∈ italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT × italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT and a corresponding mild solution (T,Y)∈C t 0⁢L x∞×C t 0⁢L x∞𝑇 𝑌 subscript superscript 𝐶 0 𝑡 subscript superscript 𝐿 𝑥 subscript superscript 𝐶 0 𝑡 subscript superscript 𝐿 𝑥(T,Y)\in C^{0}{t}L^{\infty}{x}\times C^{0}{t}L^{\infty}{x}( italic_T , italic_Y ) ∈ italic_C start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT × italic_C start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT of (1.2), then

‖T⁢(x,t)‖L x∞≤e−λ⁢t⁢‖T 0⁢(x)‖L x∞+λ−1⁢‖Y 0⁢(x)‖L x∞.subscript norm 𝑇 𝑥 𝑡 subscript superscript 𝐿 𝑥 superscript 𝑒 𝜆 𝑡 subscript norm subscript 𝑇 0 𝑥 subscript superscript 𝐿 𝑥 superscript 𝜆 1 subscript norm subscript 𝑌 0 𝑥 subscript superscript 𝐿 𝑥\left|T(x,t)\right|{L^{\infty}{x}}\leq e^{-\lambda t}\left|T_{0}(x)\right% |{L^{\infty}{x}}+\lambda^{-1}\left|Y_{0}(x)\right|{L^{\infty}{x}}.∥ italic_T ( italic_x , italic_t ) ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT ≤ italic_e start_POSTSUPERSCRIPT - italic_λ italic_t end_POSTSUPERSCRIPT ∥ italic_T start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_x ) ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT + italic_λ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ∥ italic_Y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_x ) ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT .(2.5)

Proof.

Using lemma 2.1 and |r⁢(T)|≤1 𝑟 𝑇 1\left|r(T)\right|\leq 1| italic_r ( italic_T ) | ≤ 1, we have

‖T‖L x∞subscript norm 𝑇 subscript superscript 𝐿 𝑥\displaystyle\left|T\right|{L^{\infty}{x}}∥ italic_T ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT≤e−λ⁢t⁢‖T 0‖L x∞+sup t>0‖Y⁢r⁢(T)‖L x∞⁢∫0 t e−λ⁢(t−τ)⁢d τ≤e−λ⁢t⁢‖T 0‖L x∞+λ−1⁢(1−e−λ⁢t)⁢‖Y 0‖L x∞.absent superscript 𝑒 𝜆 𝑡 subscript norm subscript 𝑇 0 subscript superscript 𝐿 𝑥 subscript supremum 𝑡 0 subscript norm 𝑌 𝑟 𝑇 subscript superscript 𝐿 𝑥 superscript subscript 0 𝑡 superscript 𝑒 𝜆 𝑡 𝜏 differential-d 𝜏 superscript 𝑒 𝜆 𝑡 subscript norm subscript 𝑇 0 subscript superscript 𝐿 𝑥 superscript 𝜆 1 1 superscript 𝑒 𝜆 𝑡 subscript norm subscript 𝑌 0 subscript superscript 𝐿 𝑥\displaystyle\leq e^{-\lambda t}\left|T_{0}\right|{L^{\infty}{x}}+\sup_{t>% 0}\left|Yr\left(T\right)\right|{L^{\infty}{x}}\int_{0}^{t}e^{-\lambda\left% (t-\tau\right)}\ \mathrm{d}\tau\leq e^{-\lambda t}\left|T_{0}\right|{L^{% \infty}{x}}+\lambda^{-1}\left(1-e^{-\lambda t}\right)\left|Y_{0}\right|{L^% {\infty}{x}}.≤ italic_e start_POSTSUPERSCRIPT - italic_λ italic_t end_POSTSUPERSCRIPT ∥ italic_T start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT + roman_sup start_POSTSUBSCRIPT italic_t > 0 end_POSTSUBSCRIPT ∥ italic_Y italic_r ( italic_T ) ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT - italic_λ ( italic_t - italic_τ ) end_POSTSUPERSCRIPT roman_d italic_τ ≤ italic_e start_POSTSUPERSCRIPT - italic_λ italic_t end_POSTSUPERSCRIPT ∥ italic_T start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT + italic_λ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( 1 - italic_e start_POSTSUPERSCRIPT - italic_λ italic_t end_POSTSUPERSCRIPT ) ∥ italic_Y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT .

∎

Proposition 2.6.

Suppose λ>0 𝜆 0\lambda>0 italic_λ > 0. If we are given an initial datum (T 0,Y 0)∈(L x 2∩L x∞)×L x∞subscript 𝑇 0 subscript 𝑌 0 subscript superscript 𝐿 2 𝑥 subscript superscript 𝐿 𝑥 subscript superscript 𝐿 𝑥(T_{0},Y_{0})\in\left(L^{2}{x}\cap L^{\infty}{x}\right)\times L^{\infty}{x}( italic_T start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_Y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) ∈ ( italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ∩ italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ) × italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT and a corresponding mild solution (T,Y)∈C t 0⁢(L x 2∩L x∞)×C t 0⁢L x∞𝑇 𝑌 subscript superscript 𝐶 0 𝑡 subscript superscript 𝐿 2 𝑥 subscript superscript 𝐿 𝑥 subscript superscript 𝐶 0 𝑡 subscript superscript 𝐿 𝑥(T,Y)\in C^{0}{t}\left(L^{2}{x}\cap L^{\infty}{x}\right)\times C^{0}{t}L^{% \infty}{x}( italic_T , italic_Y ) ∈ italic_C start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ( italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ∩ italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ) × italic_C start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT of (1.2), then

‖T‖L x 2 2≲min⁡{‖T 0‖L x 2 2+C⁢(‖T 0‖L x∞,‖Y 0‖L x∞,λ,β)⁢‖Y 0−Y‖L x 1,‖T 0‖L x 2 2⁢e(‖Y 0‖L x∞−λ)⁢t}less-than-or-similar-to superscript subscript norm 𝑇 subscript superscript 𝐿 2 𝑥 2 superscript subscript norm subscript 𝑇 0 subscript superscript 𝐿 2 𝑥 2 𝐶 subscript norm subscript 𝑇 0 subscript superscript 𝐿 𝑥 subscript norm subscript 𝑌 0 subscript superscript 𝐿 𝑥 𝜆 𝛽 subscript norm subscript 𝑌 0 𝑌 subscript superscript 𝐿 1 𝑥 superscript subscript norm subscript 𝑇 0 subscript superscript 𝐿 2 𝑥 2 superscript 𝑒 subscript norm subscript 𝑌 0 subscript superscript 𝐿 𝑥 𝜆 𝑡\left|T\right|{L^{2}{x}}^{2}\lesssim\min\left{\left|T_{0}\right|{L^{2}% {x}}^{2}+C\left(\left|T{0}\right|{L^{\infty}{x}},\left|Y{0}\right|{L% ^{\infty}{x}},\lambda,\beta\right)\left|Y_{0}-Y\right|{L^{1}{x}},\ \left% |T_{0}\right|{L^{2}{x}}^{2}e^{\left(\left|Y_{0}\right|{L^{\infty}{x}}-% \lambda\right)t}\right}∥ italic_T ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≲ roman_min { ∥ italic_T start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_C ( ∥ italic_T start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT , ∥ italic_Y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT , italic_λ , italic_β ) ∥ italic_Y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT - italic_Y ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT , ∥ italic_T start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT ( ∥ italic_Y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT - italic_λ ) italic_t end_POSTSUPERSCRIPT }(2.6)

In particular, if (T 0,Y 0)∈(L x 2∩L x∞)×(L x 1∩L x∞)subscript 𝑇 0 subscript 𝑌 0 subscript superscript 𝐿 2 𝑥 subscript superscript 𝐿 𝑥 subscript superscript 𝐿 1 𝑥 subscript superscript 𝐿 𝑥(T_{0},Y_{0})\in\left(L^{2}{x}\cap L^{\infty}{x}\right)\times\left(L^{1}{x}% \cap L^{\infty}{x}\right)( italic_T start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_Y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) ∈ ( italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ∩ italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ) × ( italic_L start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ∩ italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ) then

sup t>0‖T⁢(x,t)‖L x 2≤C⁢(‖T 0‖L x 2∩L x∞,‖Y 0‖L x 1∩L x∞,λ,β).subscript supremum 𝑡 0 subscript norm 𝑇 𝑥 𝑡 subscript superscript 𝐿 2 𝑥 𝐶 subscript norm subscript 𝑇 0 subscript superscript 𝐿 2 𝑥 subscript superscript 𝐿 𝑥 subscript norm subscript 𝑌 0 subscript superscript 𝐿 1 𝑥 subscript superscript 𝐿 𝑥 𝜆 𝛽\sup_{t>0}\left|T(x,t)\right|{L^{2}{x}}\leq C\left(\left|T_{0}\right|{L% ^{2}{x}\cap L^{\infty}{x}},\left|Y{0}\right|{L^{1}{x}\cap L^{\infty}_{x% }},\lambda,\beta\right).roman_sup start_POSTSUBSCRIPT italic_t > 0 end_POSTSUBSCRIPT ∥ italic_T ( italic_x , italic_t ) ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT ≤ italic_C ( ∥ italic_T start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ∩ italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT , ∥ italic_Y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ∩ italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT , italic_λ , italic_β ) .(2.7)

Proof.

Recall that T⁢(x,t)𝑇 𝑥 𝑡 T(x,t)italic_T ( italic_x , italic_t ) is smooth in x 𝑥 x italic_x, so we can write T t=(Δ−λ)⁢T+Y⁢r⁢(T)subscript 𝑇 𝑡 Δ 𝜆 𝑇 𝑌 𝑟 𝑇 T_{t}=\left(\Delta-\lambda\right)T+Yr\left(T\right)italic_T start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT = ( roman_Δ - italic_λ ) italic_T + italic_Y italic_r ( italic_T ) where equality is understood in an L x 2 subscript superscript 𝐿 2 𝑥 L^{2}_{x}italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT sense. We then use a standard energy argument:

d d⁢t⁢1 2⁢‖T‖L x 2 2 d d 𝑡 1 2 superscript subscript norm 𝑇 subscript superscript 𝐿 2 𝑥 2\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\frac{1}{2}\left|T\right|{L^{2}% {x}}^{2}divide start_ARG roman_d end_ARG start_ARG roman_d italic_t end_ARG divide start_ARG 1 end_ARG start_ARG 2 end_ARG ∥ italic_T ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT=−‖T‖H˙x 1 2−λ 2⁢‖T‖L x 2 2+∫T⁢(x,t)⁢Y⁢(x,t)⁢r⁢(T⁢(x,t))⁢d x absent subscript superscript norm 𝑇 2 subscript superscript˙𝐻 1 𝑥 𝜆 2 subscript superscript norm 𝑇 2 subscript superscript 𝐿 2 𝑥 𝑇 𝑥 𝑡 𝑌 𝑥 𝑡 𝑟 𝑇 𝑥 𝑡 differential-d 𝑥\displaystyle=-\left|T\right|^{2}{\dot{H}^{1}{x}}-\frac{\lambda}{2}\left|% T\right|^{2}{L^{2}{x}}+\int T(x,t)\ Y(x,t)\ r\left(T(x,t)\right)\ \mathrm{d}x= - ∥ italic_T ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT over˙ start_ARG italic_H end_ARG start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT - divide start_ARG italic_λ end_ARG start_ARG 2 end_ARG ∥ italic_T ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT + ∫ italic_T ( italic_x , italic_t ) italic_Y ( italic_x , italic_t ) italic_r ( italic_T ( italic_x , italic_t ) ) roman_d italic_x ≤∫T⁢(x,t)⁢Y⁢(x,t)⁢r⁢(T⁢(x,t))⁢d x absent 𝑇 𝑥 𝑡 𝑌 𝑥 𝑡 𝑟 𝑇 𝑥 𝑡 differential-d 𝑥\displaystyle\leq\int T(x,t)\ Y(x,t)\ r\left(T(x,t)\right)\ \mathrm{d}x≤ ∫ italic_T ( italic_x , italic_t ) italic_Y ( italic_x , italic_t ) italic_r ( italic_T ( italic_x , italic_t ) ) roman_d italic_x(2.8) =β−1⁢∫T⁢(x,t)⁢(−Y t⁢(x,t))⁢d x absent superscript 𝛽 1 𝑇 𝑥 𝑡 subscript 𝑌 𝑡 𝑥 𝑡 differential-d 𝑥\displaystyle=\beta^{-1}\int T(x,t)\ \left(-Y_{t}(x,t)\right)\ \mathrm{d}x= italic_β start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ∫ italic_T ( italic_x , italic_t ) ( - italic_Y start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ( italic_x , italic_t ) ) roman_d italic_x ≤β−1⁢sup t>0‖T⁢(x,t)‖L x∞⁢∫(−Y t⁢(x,t))⁢d x.absent superscript 𝛽 1 subscript supremum 𝑡 0 subscript norm 𝑇 𝑥 𝑡 subscript superscript 𝐿 𝑥 subscript 𝑌 𝑡 𝑥 𝑡 differential-d 𝑥\displaystyle\leq\beta^{-1}\sup_{t>0}\left|T(x,t)\right|{L^{\infty}{x}}% \int\left(-Y_{t}(x,t)\right)\ \mathrm{d}x.≤ italic_β start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT roman_sup start_POSTSUBSCRIPT italic_t > 0 end_POSTSUBSCRIPT ∥ italic_T ( italic_x , italic_t ) ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∫ ( - italic_Y start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ( italic_x , italic_t ) ) roman_d italic_x .

Integrating both sides of the above over [0,t]0 𝑡[0,t][ 0 , italic_t ] and using proposition 2.5 gives

‖T⁢(x,t)‖L x 2 2≲‖T 0⁢(x)‖L x 2 2+C⁢(‖T 0‖L x∞,‖Y 0‖L x∞,λ,β)⁢∫0 t∫(−Y τ⁢(x,τ))⁢d x⁢d τ.less-than-or-similar-to superscript subscript norm 𝑇 𝑥 𝑡 subscript superscript 𝐿 2 𝑥 2 superscript subscript norm subscript 𝑇 0 𝑥 subscript superscript 𝐿 2 𝑥 2 𝐶 subscript norm subscript 𝑇 0 subscript superscript 𝐿 𝑥 subscript norm subscript 𝑌 0 subscript superscript 𝐿 𝑥 𝜆 𝛽 superscript subscript 0 𝑡 subscript 𝑌 𝜏 𝑥 𝜏 differential-d 𝑥 differential-d 𝜏\displaystyle\left|T(x,t)\right|{L^{2}{x}}^{2}\lesssim\left|T_{0}(x)% \right|{L^{2}{x}}^{2}+C\left(\left|T_{0}\right|{L^{\infty}{x}},\left|Y% {0}\right|{L^{\infty}{x}},\lambda,\beta\right)\int{0}^{t}\int\left(-Y_{% \tau}(x,\tau)\right)\ \mathrm{d}x\ \mathrm{d}\tau.∥ italic_T ( italic_x , italic_t ) ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≲ ∥ italic_T start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_x ) ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_C ( ∥ italic_T start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT , ∥ italic_Y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT , italic_λ , italic_β ) ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ∫ ( - italic_Y start_POSTSUBSCRIPT italic_τ end_POSTSUBSCRIPT ( italic_x , italic_τ ) ) roman_d italic_x roman_d italic_τ .

Then, we use Fubini’s theorem and the fundamental theorem of calculus to obtain one part of (2.6). The special case (2.7) follows since Y 0⁢(x)−Y⁢(x,t)≤Y 0⁢(x)subscript 𝑌 0 𝑥 𝑌 𝑥 𝑡 subscript 𝑌 0 𝑥 Y_{0}(x)-Y(x,t)\leq Y_{0}(x)italic_Y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_x ) - italic_Y ( italic_x , italic_t ) ≤ italic_Y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_x ). For the other part of (2.6), we keep the λ 𝜆\lambda italic_λ term around, instead use T⁢r⁢(T)≲T 2 less-than-or-similar-to 𝑇 𝑟 𝑇 superscript 𝑇 2 Tr(T)\lesssim T^{2}italic_T italic_r ( italic_T ) ≲ italic_T start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT in (2.8), place Y 𝑌 Y italic_Y in L x∞subscript superscript 𝐿 𝑥 L^{\infty}_{x}italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT, and conclude with an application of Grönwall’s inequality. ∎

Next, I demonstrate how these a priori estimates may be applied to obtain global-in-time mild solutions to (1.2).

Corollary 2.7.

Suppose λ>0 𝜆 0\lambda>0 italic_λ > 0 and we are given an initial datum (T 0,Y 0)∈L x∞×L x∞subscript 𝑇 0 subscript 𝑌 0 subscript superscript 𝐿 𝑥 subscript superscript 𝐿 𝑥\left(T_{0},Y_{0}\right)\in L^{\infty}{x}\times L^{\infty}{x}( italic_T start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_Y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) ∈ italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT × italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT. Then, (1.2) admits a unique mild solution (T,Y)∈C t 0⁢L x∞×C t 0⁢L x∞𝑇 𝑌 subscript superscript 𝐶 0 𝑡 subscript superscript 𝐿 𝑥 subscript superscript 𝐶 0 𝑡 subscript superscript 𝐿 𝑥(T,Y)\in C^{0}{t}L^{\infty}{x}\times C^{0}{t}L^{\infty}{x}( italic_T , italic_Y ) ∈ italic_C start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT × italic_C start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT valid for t∈[0,∞)𝑡 0 t\in[0,\infty)italic_t ∈ [ 0 , ∞ ). Further, if we also have T 0∈L x 2 subscript 𝑇 0 subscript superscript 𝐿 2 𝑥 T_{0}\in L^{2}_{x}italic_T start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT and Y 0∈L x 1 subscript 𝑌 0 subscript superscript 𝐿 1 𝑥 Y_{0}\in L^{1}_{x}italic_Y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ italic_L start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT, then the mild solution obeys (2.7).

Proof.

Fixed-point iteration (see for example [16]) can be used to established local-in-time existence and uniqueness. Proposition 2.5 allows us to extend this local solution to a global one. With this solution in hand, we apply proposition 2.6 to conclude. ∎

Consequently, solutions to (1.2) do not exhibit thermal blow-up when λ>0 𝜆 0\lambda>0 italic_λ > 0. In the case λ=0 𝜆 0\lambda=0 italic_λ = 0 the techniques from proposition 2.5 would yield

‖T⁢(x,t)‖L x∞≤‖T 0⁢(x)‖L x∞+t⁢‖Y 0⁢(x)‖L x∞.subscript norm 𝑇 𝑥 𝑡 subscript superscript 𝐿 𝑥 subscript norm subscript 𝑇 0 𝑥 subscript superscript 𝐿 𝑥 𝑡 subscript norm subscript 𝑌 0 𝑥 subscript superscript 𝐿 𝑥\left|T(x,t)\right|{L^{\infty}{x}}\leq\left|T_{0}(x)\right|{L^{\infty}% {x}}+t\left|Y_{0}(x)\right|{L^{\infty}{x}}.∥ italic_T ( italic_x , italic_t ) ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT ≤ ∥ italic_T start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_x ) ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT + italic_t ∥ italic_Y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_x ) ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT .

While this does not give a nice uniform-in-time bound, it at least says that thermal blow-up does not occur in finite time irrespective of λ 𝜆\lambda italic_λ.

Even though the mild solutions constructed above are smooth with respect to x 𝑥 x italic_x, they are not necessarily classical solutions (that is, they are not also continuously differentiable in t 𝑡 t italic_t). Indeed, the PDE T t=Δ⁢T+Y⁢r⁢(T)subscript 𝑇 𝑡 Δ 𝑇 𝑌 𝑟 𝑇 T_{t}=\Delta T+Yr(T)italic_T start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT = roman_Δ italic_T + italic_Y italic_r ( italic_T ) tells us that T t subscript 𝑇 𝑡 T_{t}italic_T start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT only has the spatial regularity of Y 𝑌 Y italic_Y, itself controlled by the regularity of Y 0 subscript 𝑌 0 Y_{0}italic_Y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT. However, if Y 0 subscript 𝑌 0 Y_{0}italic_Y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT is continuous, then so is T t subscript 𝑇 𝑡 T_{t}italic_T start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT, and we indeed have a classical solution. In particular, the choice Y 0≡1 subscript 𝑌 0 1 Y_{0}\equiv 1 italic_Y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ≡ 1 used in [11] gives rise to a unique global classical solution.

For the sake of completeness, I mention that we can obtain coarse estimates on ∇T∇𝑇\nabla T∇ italic_T without too much trouble:

Proposition 2.8.

Pick p,q,ℓ∈[1,∞]𝑝 𝑞 ℓ 1 p,q,\ell\in\left[1,\infty\right]italic_p , italic_q , roman_ℓ ∈ [ 1 , ∞ ] with p−1=q−1−ℓ−1 superscript 𝑝 1 superscript 𝑞 1 superscript ℓ 1 p^{-1}=q^{-1}-\ell^{-1}italic_p start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT = italic_q start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT - roman_ℓ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT and ℓ>d ℓ 𝑑\ell>d roman_ℓ > italic_d. If (T 0,Y 0)∈(L x p∩L x∞)×(L x q∩L x∞)subscript 𝑇 0 subscript 𝑌 0 subscript superscript 𝐿 𝑝 𝑥 subscript superscript 𝐿 𝑥 subscript superscript 𝐿 𝑞 𝑥 subscript superscript 𝐿 𝑥(T_{0},Y_{0})\in\left(L^{p}{x}\cap L^{\infty}{x}\right)\times\left(L^{q}{x}% \cap L^{\infty}{x}\right)( italic_T start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_Y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) ∈ ( italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ∩ italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ) × ( italic_L start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ∩ italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ) then the corresponding mild solution (T,Y)𝑇 𝑌(T,Y)( italic_T , italic_Y ) of (1.2) satisfies

‖∇T⁢(x,t)‖L x p≲t−1 2⁢‖T 0⁢(x)‖L x p+t 1 2⁢(1−d ℓ)⁢‖Y 0‖L x q⁢∀t>0.less-than-or-similar-to subscript norm∇𝑇 𝑥 𝑡 subscript superscript 𝐿 𝑝 𝑥 superscript 𝑡 1 2 subscript norm subscript 𝑇 0 𝑥 subscript superscript 𝐿 𝑝 𝑥 superscript 𝑡 1 2 1 𝑑 ℓ subscript norm subscript 𝑌 0 subscript superscript 𝐿 𝑞 𝑥 for-all 𝑡 0\left|\nabla T(x,t)\right|{L^{p}{x}}\lesssim t^{-\frac{1}{2}}\left|T_{0}(% x)\right|{L^{p}{x}}+t^{\frac{1}{2}\left(1-\frac{d}{\ell}\right)}\left|Y_{0% }\right|{L^{q}{x}}\quad\forall\ t>0.∥ ∇ italic_T ( italic_x , italic_t ) ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT ≲ italic_t start_POSTSUPERSCRIPT - divide start_ARG 1 end_ARG start_ARG 2 end_ARG end_POSTSUPERSCRIPT ∥ italic_T start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_x ) ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT + italic_t start_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( 1 - divide start_ARG italic_d end_ARG start_ARG roman_ℓ end_ARG ) end_POSTSUPERSCRIPT ∥ italic_Y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∀ italic_t > 0 .(2.9)

In particular,

‖∇T⁢(x,t)‖L x∞≲t−1 2⁢‖T 0⁢(x)‖L x∞+t 1 2⁢‖Y 0‖L x∞⁢∀t>0.less-than-or-similar-to subscript norm∇𝑇 𝑥 𝑡 subscript superscript 𝐿 𝑥 superscript 𝑡 1 2 subscript norm subscript 𝑇 0 𝑥 subscript superscript 𝐿 𝑥 superscript 𝑡 1 2 subscript norm subscript 𝑌 0 subscript superscript 𝐿 𝑥 for-all 𝑡 0\left|\nabla T(x,t)\right|{L^{\infty}{x}}\lesssim t^{-\frac{1}{2}}\left|T% {0}(x)\right|{L^{\infty}{x}}+t^{\frac{1}{2}}\left|Y{0}\right|{L^{% \infty}{x}}\quad\forall\ t>0.∥ ∇ italic_T ( italic_x , italic_t ) ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT ≲ italic_t start_POSTSUPERSCRIPT - divide start_ARG 1 end_ARG start_ARG 2 end_ARG end_POSTSUPERSCRIPT ∥ italic_T start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_x ) ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT + italic_t start_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG 2 end_ARG end_POSTSUPERSCRIPT ∥ italic_Y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∀ italic_t > 0 .(2.10)

Proof.

Using (2.2) and |r⁢(T)|≤1 𝑟 𝑇 1|r(T)|\leq 1| italic_r ( italic_T ) | ≤ 1, we have

‖∇T⁢(x,t)‖L x p subscript norm∇𝑇 𝑥 𝑡 subscript superscript 𝐿 𝑝 𝑥\displaystyle\left|\nabla T(x,t)\right|{L^{p}{x}}∥ ∇ italic_T ( italic_x , italic_t ) ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT≲t−1 2⁢‖T 0‖L x p+‖Y 0‖L x q⁢∫0 t(t−τ)−1 2⁢(d ℓ+1)⁢d τ less-than-or-similar-to absent superscript 𝑡 1 2 subscript norm subscript 𝑇 0 subscript superscript 𝐿 𝑝 𝑥 subscript norm subscript 𝑌 0 subscript superscript 𝐿 𝑞 𝑥 superscript subscript 0 𝑡 superscript 𝑡 𝜏 1 2 𝑑 ℓ 1 differential-d 𝜏\displaystyle\lesssim t^{-\frac{1}{2}}\left|T_{0}\right|{L^{p}{x}}+\left|% Y_{0}\right|{L^{q}{x}}\int_{0}^{t}\left(t-\tau\right)^{-\frac{1}{2}\left(% \frac{d}{\ell}+1\right)}\ \mathrm{d}\tau≲ italic_t start_POSTSUPERSCRIPT - divide start_ARG 1 end_ARG start_ARG 2 end_ARG end_POSTSUPERSCRIPT ∥ italic_T start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT + ∥ italic_Y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ( italic_t - italic_τ ) start_POSTSUPERSCRIPT - divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( divide start_ARG italic_d end_ARG start_ARG roman_ℓ end_ARG + 1 ) end_POSTSUPERSCRIPT roman_d italic_τ

and we’re all done upon computing the integral. ∎

Thus solutions of (1.2) do not develop sharp edges in finite time. I expect that one can get rid of the temporally growing term in (2.9) using a more thorough analysis.

2.4 Decay of Small-Data Solutions

Next I show that, if the initial temperature T 0 subscript 𝑇 0 T_{0}italic_T start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT is sufficiently small, then lim t→∞|T⁢(x,t)|=0 subscript→𝑡 𝑇 𝑥 𝑡 0\lim_{t\rightarrow\infty}|T(x,t)|=0 roman_lim start_POSTSUBSCRIPT italic_t → ∞ end_POSTSUBSCRIPT | italic_T ( italic_x , italic_t ) | = 0 uniformly. This confirms that (1.2) has a nonzero auto-ignition temperature up to some mild assumptions on T 0 subscript 𝑇 0 T_{0}italic_T start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT. From our work in the last section, we know that λ=0 𝜆 0\lambda=0 italic_λ = 0 gives the worst L x∞subscript superscript 𝐿 𝑥 L^{\infty}_{x}italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT temperature bounds possible, so in this section we’ll set λ=0 𝜆 0\lambda=0 italic_λ = 0 for concreteness: if we can establish decay with λ=0 𝜆 0\lambda=0 italic_λ = 0 then it is even simpler to establish decay with λ>0 𝜆 0\lambda>0 italic_λ > 0. While this means we don’t have access to the existence-uniqueness results from the previous section, we can still build a temporally-decaying global solution using the assumption of small initial data and the bootstrap principle. This strategy for establishing time decay is well-known in the theory of dispersive PDEs, see for instance [13].

Throughout this section, we suppose that ϵ 0∈(0,1)subscript italic-ϵ 0 0 1\epsilon_{0}\in(0,1)italic_ϵ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ ( 0 , 1 ) is a small parameter (to be constrained later) and

‖T 0‖L x 1∩L x∞⁢<ϵ 0⁢and∥⁢Y 0∥L x∞<∞.evaluated-at subscript norm subscript 𝑇 0 subscript superscript 𝐿 1 𝑥 subscript superscript 𝐿 𝑥 bra subscript italic-ϵ 0 and subscript 𝑌 0 subscript superscript 𝐿 𝑥\left|T_{0}\right|{L^{1}{x}\cap L^{\infty}{x}}<\epsilon{0}\quad\text{and% }\quad\left|Y_{0}\right|{L^{\infty}{x}}<\infty.∥ italic_T start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ∩ italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT < italic_ϵ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and ∥ italic_Y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT < ∞ .(2.11)

The requirement that T 0 subscript 𝑇 0 T_{0}italic_T start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT is small in L x 1 subscript superscript 𝐿 1 𝑥 L^{1}{x}italic_L start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT is a minor technical one giving us access to a decay estimate. Now, given t∗∈(0,∞]subscript 𝑡 0 t{}\in(0,\infty]italic_t start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ∈ ( 0 , ∞ ] and a sufficiently nice function T⁢(x,t)𝑇 𝑥 𝑡 T(x,t)italic_T ( italic_x , italic_t ) defined for t∈[0,t∗]𝑡 0 subscript 𝑡 t\in[0,t_{}]italic_t ∈ [ 0 , italic_t start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ], we set

‖T⁢(x,t)‖X t∗≐sup t∈[0,t∗][(1+t)d 2⁢‖T⁢(x,t)‖L x∞+‖T⁢(x,t)‖L x 1].approaches-limit subscript norm 𝑇 𝑥 𝑡 subscript 𝑋 subscript 𝑡 subscript supremum 𝑡 0 subscript 𝑡 delimited-[]superscript 1 𝑡 𝑑 2 subscript norm 𝑇 𝑥 𝑡 subscript superscript 𝐿 𝑥 subscript norm 𝑇 𝑥 𝑡 subscript superscript 𝐿 1 𝑥\left|T(x,t)\right|{X{t_{}}}\doteq\sup_{t\in[0,t_{}]}\left[\left(1+t% \right)^{\frac{d}{2}}\left|T(x,t)\right|{L^{\infty}{x}}+\left|T(x,t)% \right|{L^{1}{x}}\right].∥ italic_T ( italic_x , italic_t ) ∥ start_POSTSUBSCRIPT italic_X start_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT ≐ roman_sup start_POSTSUBSCRIPT italic_t ∈ [ 0 , italic_t start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ] end_POSTSUBSCRIPT [ ( 1 + italic_t ) start_POSTSUPERSCRIPT divide start_ARG italic_d end_ARG start_ARG 2 end_ARG end_POSTSUPERSCRIPT ∥ italic_T ( italic_x , italic_t ) ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT + ∥ italic_T ( italic_x , italic_t ) ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT ] .

This norm naturally gives rise to a Banach space X t∗subscript 𝑋 subscript 𝑡 X_{t_{*}}italic_X start_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT end_POSTSUBSCRIPT defined by

X t∗≐{T:[0,t∗]→L x 1∩L x∞continuous|∥T(x,t)∥X t∗<∞}.X_{t_{}}\doteq\left{T\colon[0,t_{}]\rightarrow L^{1}{x}\cap L^{\infty}{x}% \ \text{continuous}\ \bigg{|}\ \left|T(x,t)\right|{X{t_{*}}}<\infty\right}.italic_X start_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT end_POSTSUBSCRIPT ≐ { italic_T : [ 0 , italic_t start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ] → italic_L start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ∩ italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT continuous | ∥ italic_T ( italic_x , italic_t ) ∥ start_POSTSUBSCRIPT italic_X start_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT < ∞ } .

The first order of business is to prove a new a priori estimate for solutions that are small in X t∗subscript 𝑋 subscript 𝑡 X_{t_{*}}italic_X start_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT end_POSTSUBSCRIPT.

Proposition 2.9.

Take ϵ 1∈[ϵ 0,1)subscript italic-ϵ 1 subscript italic-ϵ 0 1\epsilon_{1}\in\left[\epsilon_{0},1\right)italic_ϵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∈ [ italic_ϵ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , 1 ). Assume that (2.11) holds and that there is a corresponding mild solution (T,Y)∈X t∗×C t 0⁢L x∞𝑇 𝑌 subscript 𝑋 subscript 𝑡 subscript superscript 𝐶 0 𝑡 subscript superscript 𝐿 𝑥\left(T,Y\right)\in X_{t_{}}\times C^{0}{t}L^{\infty}{x}( italic_T , italic_Y ) ∈ italic_X start_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT end_POSTSUBSCRIPT × italic_C start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT to (1.2) valid on [0,t∗]0 subscript 𝑡[0,t_{}][ 0 , italic_t start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ]. Further, assume that ‖T‖X t∗≤ϵ 1 subscript norm 𝑇 subscript 𝑋 subscript 𝑡 subscript italic-ϵ 1\left|T\right|{X{t_{*}}}\leq\epsilon_{1}∥ italic_T ∥ start_POSTSUBSCRIPT italic_X start_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT ≤ italic_ϵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT. Then, there exists an absolute constant C>0 𝐶 0 C>0 italic_C > 0 such that

‖T⁢(x,t)‖X t∗≤2⁢ϵ 0+C⁢‖Y 0‖L x∞⁢ϵ 1 11.subscript norm 𝑇 𝑥 𝑡 subscript 𝑋 subscript 𝑡 2 subscript italic-ϵ 0 𝐶 subscript norm subscript 𝑌 0 subscript superscript 𝐿 𝑥 superscript subscript italic-ϵ 1 11\left|T(x,t)\right|{X{t_{*}}}\leq 2\epsilon_{0}+C\left|Y_{0}\right|{L^{% \infty}{x}}\epsilon_{1}^{11}.∥ italic_T ( italic_x , italic_t ) ∥ start_POSTSUBSCRIPT italic_X start_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT ≤ 2 italic_ϵ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_C ∥ italic_Y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_ϵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 11 end_POSTSUPERSCRIPT .(2.12)

Proof.

Let’s look at the L x∞subscript superscript 𝐿 𝑥 L^{\infty}{x}italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT part of the X t∗subscript 𝑋 subscript 𝑡 X{t_{*}}italic_X start_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT end_POSTSUBSCRIPT norm first. Together with the hypothesis of our claim, lemma 2.2 (with p=11 𝑝 11 p=11 italic_p = 11) implies ‖r⁢(T)‖L x∞≤C⁢ϵ 1 11⁢(1+t)−11 2⁢d subscript norm 𝑟 𝑇 subscript superscript 𝐿 𝑥 𝐶 superscript subscript italic-ϵ 1 11 superscript 1 𝑡 11 2 𝑑\left|r(T)\right|{L^{\infty}{x}}\leq C\epsilon_{1}^{11}\left(1+t\right)^{-% \frac{11}{2}d}∥ italic_r ( italic_T ) ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT ≤ italic_C italic_ϵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 11 end_POSTSUPERSCRIPT ( 1 + italic_t ) start_POSTSUPERSCRIPT - divide start_ARG 11 end_ARG start_ARG 2 end_ARG italic_d end_POSTSUPERSCRIPT and ‖r⁢(T)‖L x 1≤C⁢ϵ 1 11⁢(1+t)−5⁢d subscript norm 𝑟 𝑇 subscript superscript 𝐿 1 𝑥 𝐶 superscript subscript italic-ϵ 1 11 superscript 1 𝑡 5 𝑑\left|r(T)\right|{L^{1}{x}}\leq C\epsilon_{1}^{11}\left(1+t\right)^{-5d}∥ italic_r ( italic_T ) ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT ≤ italic_C italic_ϵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 11 end_POSTSUPERSCRIPT ( 1 + italic_t ) start_POSTSUPERSCRIPT - 5 italic_d end_POSTSUPERSCRIPT. Combining these estimates with (2.1), (2.2), and (2.11), we have for t>1 𝑡 1 t>1 italic_t > 1

‖T⁢(x,t)‖L x∞subscript norm 𝑇 𝑥 𝑡 subscript superscript 𝐿 𝑥\displaystyle\left|T(x,t)\right|{L^{\infty}{x}}∥ italic_T ( italic_x , italic_t ) ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT≤(1+t)−d 2⁢ϵ 0+∫0 t‖e(t−τ)⁢Δ⁢Y⁢(x,τ)⁢r⁢(T⁢(x,τ))‖L x∞⁢d τ absent superscript 1 𝑡 𝑑 2 subscript italic-ϵ 0 superscript subscript 0 𝑡 subscript norm superscript 𝑒 𝑡 𝜏 Δ 𝑌 𝑥 𝜏 𝑟 𝑇 𝑥 𝜏 subscript superscript 𝐿 𝑥 differential-d 𝜏\displaystyle\leq\left(1+t\right)^{-\frac{d}{2}}\epsilon_{0}+\int_{0}^{t}\left% |e^{\left(t-\tau\right)\Delta}Y(x,\tau)r\left(T(x,\tau)\right)\right|{L^{% \infty}{x}}\ \mathrm{d}\tau≤ ( 1 + italic_t ) start_POSTSUPERSCRIPT - divide start_ARG italic_d end_ARG start_ARG 2 end_ARG end_POSTSUPERSCRIPT italic_ϵ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ∥ italic_e start_POSTSUPERSCRIPT ( italic_t - italic_τ ) roman_Δ end_POSTSUPERSCRIPT italic_Y ( italic_x , italic_τ ) italic_r ( italic_T ( italic_x , italic_τ ) ) ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT roman_d italic_τ ≤(1+t)−d 2⁢ϵ 0+∫0 t/2(t−τ)−d 2⁢‖Y⁢(x,τ)⁢r⁢(T⁢(x,τ))‖L x 1⁢d τ+∫t/2 t‖Y⁢(x,τ)⁢r⁢(T⁢(x,τ))‖L x∞⁢d τ absent superscript 1 𝑡 𝑑 2 subscript italic-ϵ 0 superscript subscript 0 𝑡 2 superscript 𝑡 𝜏 𝑑 2 subscript norm 𝑌 𝑥 𝜏 𝑟 𝑇 𝑥 𝜏 subscript superscript 𝐿 1 𝑥 differential-d 𝜏 superscript subscript 𝑡 2 𝑡 subscript norm 𝑌 𝑥 𝜏 𝑟 𝑇 𝑥 𝜏 subscript superscript 𝐿 𝑥 differential-d 𝜏\displaystyle\leq\left(1+t\right)^{-\frac{d}{2}}\epsilon_{0}+\int_{0}^{t/2}% \left(t-\tau\right)^{-\frac{d}{2}}\left|Y(x,\tau)\ r\left(T(x,\tau)\right)% \right|{L^{1}{x}}\ \mathrm{d}\tau+\int_{t/2}^{t}\left|Y(x,\tau)\ r\left(T(% x,\tau)\right)\right|{L^{\infty}{x}}\ \mathrm{d}\tau≤ ( 1 + italic_t ) start_POSTSUPERSCRIPT - divide start_ARG italic_d end_ARG start_ARG 2 end_ARG end_POSTSUPERSCRIPT italic_ϵ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t / 2 end_POSTSUPERSCRIPT ( italic_t - italic_τ ) start_POSTSUPERSCRIPT - divide start_ARG italic_d end_ARG start_ARG 2 end_ARG end_POSTSUPERSCRIPT ∥ italic_Y ( italic_x , italic_τ ) italic_r ( italic_T ( italic_x , italic_τ ) ) ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT roman_d italic_τ + ∫ start_POSTSUBSCRIPT italic_t / 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ∥ italic_Y ( italic_x , italic_τ ) italic_r ( italic_T ( italic_x , italic_τ ) ) ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT roman_d italic_τ ≤(1+t)−d 2⁢ϵ 0+C⁢‖Y 0‖L x∞⁢ϵ 1 11⁢[∫0 t/2(1+t−τ)−d 2⁢(1+τ)−5⁢d⁢d τ+∫t/2 t(1+τ)−11 2⁢d]absent superscript 1 𝑡 𝑑 2 subscript italic-ϵ 0 𝐶 subscript norm subscript 𝑌 0 subscript superscript 𝐿 𝑥 superscript subscript italic-ϵ 1 11 delimited-[]superscript subscript 0 𝑡 2 superscript 1 𝑡 𝜏 𝑑 2 superscript 1 𝜏 5 𝑑 differential-d 𝜏 superscript subscript 𝑡 2 𝑡 superscript 1 𝜏 11 2 𝑑\displaystyle\leq\left(1+t\right)^{-\frac{d}{2}}\epsilon_{0}+C\left|Y_{0}% \right|{L^{\infty}{x}}\epsilon_{1}^{11}\left[\int_{0}^{t/2}\left(1+t-\tau% \right)^{-\frac{d}{2}}\left(1+\tau\right)^{-5d}\ \mathrm{d}\tau+\int_{t/2}^{t}% \left(1+\tau\right)^{-\frac{11}{2}d}\right]≤ ( 1 + italic_t ) start_POSTSUPERSCRIPT - divide start_ARG italic_d end_ARG start_ARG 2 end_ARG end_POSTSUPERSCRIPT italic_ϵ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_C ∥ italic_Y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_ϵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 11 end_POSTSUPERSCRIPT [ ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t / 2 end_POSTSUPERSCRIPT ( 1 + italic_t - italic_τ ) start_POSTSUPERSCRIPT - divide start_ARG italic_d end_ARG start_ARG 2 end_ARG end_POSTSUPERSCRIPT ( 1 + italic_τ ) start_POSTSUPERSCRIPT - 5 italic_d end_POSTSUPERSCRIPT roman_d italic_τ + ∫ start_POSTSUBSCRIPT italic_t / 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ( 1 + italic_τ ) start_POSTSUPERSCRIPT - divide start_ARG 11 end_ARG start_ARG 2 end_ARG italic_d end_POSTSUPERSCRIPT ] ≤(1+t)−d 2⁢[ϵ 0+C⁢‖Y 0‖L x∞⁢ϵ 1 11].absent superscript 1 𝑡 𝑑 2 delimited-[]subscript italic-ϵ 0 𝐶 subscript norm subscript 𝑌 0 subscript superscript 𝐿 𝑥 superscript subscript italic-ϵ 1 11\displaystyle\leq\left(1+t\right)^{-\frac{d}{2}}\left[\epsilon_{0}+C\left|Y_{% 0}\right|{L^{\infty}{x}}\epsilon_{1}^{11}\right].≤ ( 1 + italic_t ) start_POSTSUPERSCRIPT - divide start_ARG italic_d end_ARG start_ARG 2 end_ARG end_POSTSUPERSCRIPT [ italic_ϵ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_C ∥ italic_Y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_ϵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 11 end_POSTSUPERSCRIPT ] .

For t≤1 𝑡 1 t\leq 1 italic_t ≤ 1 we may apply an almost identical approach, but we don’t split the Duhamel term and we only use (2.1). This takes care of the L x∞subscript superscript 𝐿 𝑥 L^{\infty}{x}italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT piece. The L x 1 subscript superscript 𝐿 1 𝑥 L^{1}{x}italic_L start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT estimate is obtained by a similar but even easier argument using (2.1) to control ‖e(t−τ)⁢Δ⁢Y⁢r⁢(T)‖L x 1 subscript norm superscript 𝑒 𝑡 𝜏 Δ 𝑌 𝑟 𝑇 subscript superscript 𝐿 1 𝑥\left|e^{\left(t-\tau\right)\Delta}Yr\left(T\right)\right|{L^{1}{x}}∥ italic_e start_POSTSUPERSCRIPT ( italic_t - italic_τ ) roman_Δ end_POSTSUPERSCRIPT italic_Y italic_r ( italic_T ) ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT. ∎

Note that there is nothing particularly special about the power of p=11 𝑝 11 p=11 italic_p = 11 appearing in the above proof: any large enough natural number will do, and the rapid vanishing of r⁢(T)𝑟 𝑇 r(T)italic_r ( italic_T ) at T=0 𝑇 0 T=0 italic_T = 0 guarantees we can pick such a large p 𝑝 p italic_p.

Theorem 2.10.

There exists ϵ 0=ϵ 0⁢(‖Y 0‖L x∞)∈(0,1)subscript italic-ϵ 0 subscript italic-ϵ 0 subscript norm subscript 𝑌 0 subscript superscript 𝐿 𝑥 0 1\epsilon_{0}=\epsilon_{0}\left(\left|Y_{0}\right|{L^{\infty}{x}}\right)\in% (0,1)italic_ϵ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_ϵ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( ∥ italic_Y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) ∈ ( 0 , 1 ) such that, under the hypothesis (2.11), (1.2) admits a unique global-in-time mild solution (T,Y)∈X∞×C t 0⁢L x∞𝑇 𝑌 subscript 𝑋 subscript superscript 𝐶 0 𝑡 subscript superscript 𝐿 𝑥\left(T,Y\right)\in X_{\infty}\times C^{0}{t}L^{\infty}{x}( italic_T , italic_Y ) ∈ italic_X start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT × italic_C start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT obeying the decay estimate

‖T⁢(x,t)‖L x∞≲ϵ 0⁢(1+t)−d 2.less-than-or-similar-to subscript norm 𝑇 𝑥 𝑡 subscript superscript 𝐿 𝑥 subscript italic-ϵ 0 superscript 1 𝑡 𝑑 2\left|T(x,t)\right|{L^{\infty}{x}}\lesssim\epsilon_{0}\left(1+t\right)^{-% \frac{d}{2}}.∥ italic_T ( italic_x , italic_t ) ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT ≲ italic_ϵ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( 1 + italic_t ) start_POSTSUPERSCRIPT - divide start_ARG italic_d end_ARG start_ARG 2 end_ARG end_POSTSUPERSCRIPT .(2.13)

In particular, lim t→∞|T⁢(x,t)|=0 subscript→𝑡 𝑇 𝑥 𝑡 0\lim_{t\rightarrow\infty}|T(x,t)|=0 roman_lim start_POSTSUBSCRIPT italic_t → ∞ end_POSTSUBSCRIPT | italic_T ( italic_x , italic_t ) | = 0 uniformly in x 𝑥 x italic_x.

Proof.

Using fixed-point iteration, one finds a t∗>0 subscript 𝑡 0 t_{}>0 italic_t start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT > 0 such that (1.2) has a unique local-in-time mild solution valid on [0,t∗]0 subscript 𝑡[0,t_{}][ 0 , italic_t start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ] with T⁢(x,t)∈X t∗𝑇 𝑥 𝑡 subscript 𝑋 subscript 𝑡 T(x,t)\in X_{t_{}}italic_T ( italic_x , italic_t ) ∈ italic_X start_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT end_POSTSUBSCRIPT. Further, the map τ↦‖T‖τ maps-to 𝜏 subscript norm 𝑇 𝜏\tau\mapsto\left|T\right|{\tau}italic_τ ↦ ∥ italic_T ∥ start_POSTSUBSCRIPT italic_τ end_POSTSUBSCRIPT is easily shown to be continuous for τ∈[0,t∗]𝜏 0 subscript 𝑡\tau\in[0,t{}]italic_τ ∈ [ 0 , italic_t start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ]. Accordingly, for sufficiently small t∗subscript 𝑡 t_{}italic_t start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT we know ‖T‖X t∗≤3⁢ϵ 0 subscript norm 𝑇 subscript 𝑋 subscript 𝑡 3 subscript italic-ϵ 0\left|T\right|{X{t_{}}}\leq 3\epsilon_{0}∥ italic_T ∥ start_POSTSUBSCRIPT italic_X start_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT ≤ 3 italic_ϵ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT. Now, apply proposition 2.9 with ϵ 1=3⁢ϵ 0 subscript italic-ϵ 1 3 subscript italic-ϵ 0\epsilon_{1}=3\epsilon_{0}italic_ϵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 3 italic_ϵ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT to discover

‖T‖X t∗≤2⁢ϵ 0⁢(1+C⁢‖Y 0‖L x∞⁢ϵ 0 10).subscript norm 𝑇 subscript 𝑋 subscript 𝑡 2 subscript italic-ϵ 0 1 𝐶 subscript norm subscript 𝑌 0 subscript superscript 𝐿 𝑥 superscript subscript italic-ϵ 0 10\displaystyle\left|T\right|{X{t_{*}}}\leq 2\epsilon_{0}\left(1+C\left|Y_{% 0}\right|{L^{\infty}{x}}\epsilon_{0}^{10}\right).∥ italic_T ∥ start_POSTSUBSCRIPT italic_X start_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT ≤ 2 italic_ϵ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( 1 + italic_C ∥ italic_Y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_ϵ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 10 end_POSTSUPERSCRIPT ) .

By choosing ϵ 0<1 2⁢(C⁢‖Y 0‖L x∞)−1 10 subscript italic-ϵ 0 1 2 superscript 𝐶 subscript norm subscript 𝑌 0 subscript superscript 𝐿 𝑥 1 10\epsilon_{0}<\frac{1}{2}\left(C\left|Y_{0}\right|{L^{\infty}{x}}\right)^{-% \frac{1}{10}}italic_ϵ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT < divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( italic_C ∥ italic_Y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT - divide start_ARG 1 end_ARG start_ARG 10 end_ARG end_POSTSUPERSCRIPT, the above becomes ‖T‖X t∗≤5 2⁢ϵ 0 subscript norm 𝑇 subscript 𝑋 subscript 𝑡 5 2 subscript italic-ϵ 0\left|T\right|{X{t_{*}}}\leq\frac{5}{2}\epsilon_{0}∥ italic_T ∥ start_POSTSUBSCRIPT italic_X start_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT ≤ divide start_ARG 5 end_ARG start_ARG 2 end_ARG italic_ϵ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT. The bootstrap principle (see for example [16]) allows us to iterate this argument and produce a global solution satisfying ‖T‖X∞≲ϵ 0 less-than-or-similar-to subscript norm 𝑇 subscript 𝑋 subscript italic-ϵ 0\left|T\right|{X{\infty}}\lesssim\epsilon_{0}∥ italic_T ∥ start_POSTSUBSCRIPT italic_X start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT end_POSTSUBSCRIPT ≲ italic_ϵ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT. ∎

The ϵ 0 subscript italic-ϵ 0\epsilon_{0}italic_ϵ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT from theorem 2.10 should be regarded as a lower bound on the auto-ignition temperature for the model (1.2), but my methods cannot determine how sharp this bound is.

3 Acknowledgements

I’d like to thank Carrie Clark for helpful comments on the draft manuscript, and Dominic Shillingford for invaluable discussions.

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