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Blow-up of solutions for a semilinear parabolic equation with nonlinear memory and absorption under nonlinear nonlocal boundary condition

Alexander Gladkov

Abstract

In this paper we consider initial boundary value problem for a parabolic equation with nonlinear memory and absorption under nonlinear nonlocal boundary condition. We prove global existence and blow-up of solutions.

Blow-up of solutions for a semilinear parabolic equation with nonlinear memory and absorption under nonlinear nonlocal boundary condition

Abstract

In this paper we consider initial boundary value problem for a parabolic equation with nonlinear memory and absorption under nonlinear nonlocal boundary condition. We prove global existence and blow-up of solutions.
Paper Structure (3 sections, 5 theorems, 75 equations)

This paper contains 3 sections, 5 theorems, 75 equations.

Table of Contents

  1. Introduction
  2. Blow-up in finite time
  3. Global existence

Key Result

Theorem 2.2

Let $\overline{u}$ and $\underline{u}$ be a supersolution and a subsolution of problem (v:u)--(v:n) in $Q_T,$ respectively. Suppose that $\underline{u}(x,t)> 0$ or $\overline{u}(x,t) > 0$ in ${Q}_T\cup \Gamma_T$ if $\min (q, l) < 1.$ Then $\overline{u}(x,t) \geq \underline{u}(x,t)$ in ${Q}_T\cup \Ga

Theorems & Definitions (10)

  • Definition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Lemma 2.4
  • proof
  • Theorem 2.5
  • proof
  • Theorem 3.1
  • proof
  • Remark 3.2