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Quenching phenomena in a system of non-local diffusion equations

José M. Arrieta, Raúl Ferreira, Sergio Junquera

TL;DR

This work analyzes quenching in a nonlocal diffusion system of two coupled equations with singular absorptions $u^{-p}$ and $v^{-q}$. It develops a robust well-posedness framework for a generalized integro-differential system, links global existence to stationary solutions, and characterizes the dichotomy between global existence and finite-time quenching via a parameter-dependent threshold set $U$. It then delineates when quenching is simultaneous versus non-simultaneous as a function of $p$ and $q$, and derives precise quenching-rate asymptotics for all regimes, including several borderline cases with logarithmic corrections. Numerical simulations corroborate the analytic results, illustrating stationary states, the threshold behavior, and the predicted quenching rates across parameter regimes.

Abstract

In this paper we study the quenching phenomena occurring in a non-local diffusion system of two equations with intertwined singular absorption terms of the type $u^{-p}$. We prove that there exists a range of multiplicative parameters for which every solution presents quenching, while outside this range there are both global and quenching solutions. We also characterize in terms of the exponents of the absorption terms when the quenching is simultaneous or non-simultaneous and obtain the quenching rates.

Quenching phenomena in a system of non-local diffusion equations

TL;DR

This work analyzes quenching in a nonlocal diffusion system of two coupled equations with singular absorptions and . It develops a robust well-posedness framework for a generalized integro-differential system, links global existence to stationary solutions, and characterizes the dichotomy between global existence and finite-time quenching via a parameter-dependent threshold set . It then delineates when quenching is simultaneous versus non-simultaneous as a function of and , and derives precise quenching-rate asymptotics for all regimes, including several borderline cases with logarithmic corrections. Numerical simulations corroborate the analytic results, illustrating stationary states, the threshold behavior, and the predicted quenching rates across parameter regimes.

Abstract

In this paper we study the quenching phenomena occurring in a non-local diffusion system of two equations with intertwined singular absorption terms of the type . We prove that there exists a range of multiplicative parameters for which every solution presents quenching, while outside this range there are both global and quenching solutions. We also characterize in terms of the exponents of the absorption terms when the quenching is simultaneous or non-simultaneous and obtain the quenching rates.

Paper Structure

This paper contains 8 sections, 23 theorems, 161 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Existence, uniqueness, regularity of solutions and comparison results
  3. Quenching versus global existence
  4. Simultaneous and Non-simultaneous quenching
  5. Quenching Rates
  6. Non-simultaneous quenching
  7. Simultaneous quenching
  8. Numerical simulations

Key Result

Theorem 1.1

Let $u_0$ and $v_0$ be two positive functions in $C(\overline{\Omega})$. Then there exists a unique classical solution $(u,v)$ of the problem 1.1, defined in $[0,T)$, where $T$ is the maximal existence time. Moreover, $(u,v)\in C^\infty ([0,T):C(\overline\Omega)\times C(\overline\Omega))$ and, if $T and we will say that the solution presents quenching.

Figures (8)

  • Figure 1: $p=2$ and $q=3$. To the left, $\lambda=0.001=\mu$. To the right, $\lambda=0.1,\mu = 0.001$.
  • Figure 1: $p=2$ and $q=3$. To the left, $\lambda=0.001=\mu$. To the right, $\lambda=0.1,\mu = 0.001$.
  • Figure 2: $\lambda=0.1, \mu = 0.001$, $p=2$ and $q=3$. To the left, the evolution of $(u(0,t),v(0,t))$. To the right, the simultaneous quenching rate.
  • Figure 2: $\lambda=0.1, \mu = 0.001$, $p=2$ and $q=3$. To the left, the evolution of $(u(0,t),v(0,t))$. To the right, the simultaneous quenching rate.
  • Figure 3: $\lambda=0.1=\mu$, $p=2$ and $q=0.7$. To the left, the evolution of $(u(0,t),v(0,t))$. To the right, the quenching rate.
  • ...and 3 more figures

Theorems & Definitions (46)

  • Definition 1.1
  • Theorem 1.1
  • Theorem 1.2
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 2.1
  • proof : Proof
  • ...and 36 more