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Blow-up problem for porous medium equation with absorption under nonlinear nonlocal boundary condition

Alexander Gladkov

Abstract

In this paper, we consider an initial boundary value problem for the porous medium equation with absorption under a nonlinear nonlocal boundary condition and a nonnegative initial datum. We prove the local existence of solutions, establish a comparison principle, and demonstrate both global existence and blow-up of solutions.

Blow-up problem for porous medium equation with absorption under nonlinear nonlocal boundary condition

Abstract

In this paper, we consider an initial boundary value problem for the porous medium equation with absorption under a nonlinear nonlocal boundary condition and a nonnegative initial datum. We prove the local existence of solutions, establish a comparison principle, and demonstrate both global existence and blow-up of solutions.

Paper Structure

This paper contains 5 sections, 5 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. Local existence
  3. Comparison principle
  4. Global existence
  5. Blow-up in finite time

Key Result

Lemma 2.2

There exists a sequence of positive functions $u_{0m} (x) \in L^\infty (\Omega),\,$$m \in \mathbb{N},$ possessing the following properties: For every $m \in \mathbb{N}$ there is a sequence of positive functions $u_{0mj} (x) \in C (\Omega),\,$$j \in \mathbb{N},$ satisfying the conditions

Theorems & Definitions (11)

  • Definition 2.1
  • Lemma 2.2
  • Theorem 2.3
  • proof
  • Theorem 3.1
  • proof
  • Remark 3.2
  • Theorem 4.1
  • proof
  • Theorem 5.1
  • ...and 1 more