Deriving The Upper Blow-up Rate Estimate for a Parabolic Problem

Maan A Rasheed

doi:10.24996/ijs.2020.si.1.27

Abstract

In this paper, the blow-up solutions for a parabolic problem, defined in a bounded domain, are studied. Namely, we consider the upper blow-up rate estimate for heat equation with a nonlinear Neumann boundary condition defined on a ball in Rn.

Highlights

In this paper, we study the follwoing parabolic problem:
The solutions of parabolic problems cannot be continued globally in time. This is called the blow-up phenomenon which, in time-dependent problems, has been studied over the past years by many authoes [3,4,5,6,7]. One of these problems is the problem of the heat equation defined in a ball with a nonlinear Neumann boundary condition, on which has been introduced in previous articles [7,8,9,10,11]
In an earlier study [10], it was proved that if is integrable at infinity for positive values, and is nondecreasing, with any positive initial function the blow-up occurs in a finite time

Summary

Introduction

We study the follwoing parabolic problem:. where is a ball in is the outward normal and is smooth, nonzero, nonnegative, radially symmetric, satisfiing the following condition: Many real problems in the fields of fluid dynamics, population, heat propagation, and others are modeled using the forms of parabolic partial differntional equations assosiated with different types of initial-boudary conditions [1,2,3]. The solutions of parabolic problems cannot be continued globally in time This is called the blow-up phenomenon which, in time-dependent problems, has been studied over the past years by many authoes [3,4,5,6,7]. Futhermore, in other investigations [9,11], it was proved that the upper (lower) blow-up rate estimates are as follows: Another special case, where was considered in another work [7], showing that all positive solutions blow up in finite times, and the blow-up can only occur on the boundary.

Basic properties and Blow-up Set

Conclusions