[Paper Review] Global Existence and Large Time Asymptotic Bounds of $L^{infty}$ Solutions of Thermal Diffusive Combustion Systems on $R^{n}$

Pierre Collet, Jack Xin|arXiv (Cornell University)|Dec 5, 1994

Stochastic processes and statistical mechanics9 references3 citations

TL;DR

This paper establishes the global existence of classical solutions and derives sharp large-time asymptotic bounds for the $L^∞$ norm of the temperature component in thermal-diffusive combustion systems on $\mathbb{R}^n$. Using local $L^p$ a-priori estimates and time-dependent, spatially decaying test functions, it proves that for $d > 1$ (Lewis number away from one), the $L^\infty$ norm of $u_2$ grows no faster than $O(\log \log t)$, even in the absence of a comparison principle.

ABSTRACT

We consider the initial value problem for the thermal-diffusive combustion systems of the form: $u_{1,t}= Delta_{x}u_1 - u_1 u_2^m$, $u_{2,t}= d Delta_{x} u_2 + u_1 u_2^m$, $x in R^{n}$, $n geq 1$, $m geq 1$, $d > 1$, with bounded uniformly continuous nonnegative initial data. For such initial data, solutions can be simple traveling fronts or complicated domain walls. Due to the well-known thermal-diffusive instabilities when $d$, the Lewis number, is sufficiently away from one, front solutions are potentially chaotic. It is known in the literature that solutions are uniformly bounded in time in case $d leq 1$ by a simple comparison argument. In case $d >1$, no comparison principle seems to apply. Nevertheless, we prove the existence of global classical solutions and show that the $L^{infty}$ norm of $u_2$ can not grow faster than $O(log log t)$ for any space dimension. Our main tools are local $L^{p}$ a-priori estimates and time dependent spatially decaying test functions. Our results also hold for the Arrhenius type reactions.

Motivation & Objective

• To establish global existence of classical solutions for thermal-diffusive combustion systems on $\mathbb{R}^n$ when the Lewis number $d > 1$, where standard comparison principles fail.
• To derive large-time asymptotic bounds for the $L^\infty$ norm of the temperature component $u_2$ in such systems.
• To analyze the behavior of solutions under bounded, uniformly continuous, nonnegative initial data, which may include complex structures like domain walls or chaotic fronts.
• To extend these results to Arrhenius-type reaction kinetics, broadening applicability to realistic combustion models.

Proposed method

• Employing local $L^p$ a-priori estimates to control solution growth in the absence of a comparison principle for $d > 1$.
• Introducing time-dependent, spatially decaying test functions to capture the long-time behavior and decay properties of solutions.
• Analyzing the coupled parabolic system: $u_{1,t} = \Delta_x u_1 - u_1 u_2^m$, $u_{2,t} = d \Delta_x u_2 + u_1 u_2^m$ for $x \in \mathbb{R}^n$, $n \geq 1$, $m \geq 1$, $d > 1$.
• Using energy-type estimates and weighted $L^p$ norms to derive uniform bounds on the $L^\infty$ norm of $u_2$ over time.
• Establishing that the $L^\infty$ norm of $u_2$ grows at most as $O(\log \log t)$, independent of space dimension $n$.
• Extending the analysis to Arrhenius-type reaction terms, confirming the robustness of the bounds under realistic kinetic laws.

Experimental results

Research questions

• RQ1 Can global classical solutions be proven to exist for thermal-diffusive combustion systems on $\mathbb{R}^n$ when $d > 1$, despite the absence of a comparison principle?
• RQ2 What is the optimal large-time growth rate of the $L^\infty$ norm of the temperature component $u_2$ in such systems?
• RQ3 How do time-dependent, spatially decaying test functions contribute to bounding solution norms in non-monotone parabolic systems?
• RQ4 To what extent do the derived bounds on $u_2$'s $L^\infty$ norm hold under Arrhenius-type reaction kinetics?
• RQ5 Can the $O(\log \log t)$ growth bound for $\|u_2\|_{L^\infty}$ be established uniformly across all space dimensions $n \geq 1$?

Key findings

• Global classical solutions exist for all time $t > 0$ for the thermal-diffusive combustion system on $\mathbb{R}^n$ with $d > 1$, even when no comparison principle applies.
• The $L^\infty$ norm of the temperature component $u_2$ grows no faster than $O(\log \log t)$ as $t \to \infty$, regardless of the space dimension $n \geq 1$.
• The bound $\|u_2(t)\|_{L^\infty} = O(\log \log t)$ is established through a novel use of time-dependent, spatially decaying test functions and local $L^p$ a-priori estimates.
• The results are robust and extend to Arrhenius-type reaction terms, confirming applicability to physically relevant combustion models.
• The analysis confirms that thermal-diffusive instabilities for $d > 1$ do not lead to finite-time blow-up, but rather to extremely slow growth in the $L^\infty$ norm.
• The method provides a framework to analyze long-time behavior in non-monotone parabolic systems where standard comparison techniques fail.