Blow-up of semi-discrete solution of a nonlinear parabolic equation with gradient term

TL;DR

This paper investigates the blow-up behavior of a semidiscrete approximation to a nonlinear parabolic equation with gradient terms, establishing existence, uniqueness, convergence, and blow-up rate estimates.

Contribution

It provides a rigorous analysis of blow-up phenomena in a semidiscrete setting for a nonlinear parabolic PDE with gradient terms, including convergence and blow-up rate approximation.

Findings

Semidiscrete solution exists and is unique under certain conditions.

The solution blows up in finite time.

Convergence of the semidiscrete solution to the continuous problem is proven.

Abstract

This paper is concerned with approximation of blow-up phenomena in nonlinear parabolic problems. We consider the equation u_t = u_xx +|u|^p -b(x)|u_x|^q in a bounded domain, we study the behavior of the semidiscrete problem. Under some assumptions we show existence and unicity of the semidiscrete solution, we show that it blows up in a finite time and we prove the convergence of the semidiscrete problem. Finally, we give an approximation of the blow up rate and the blow up time of the semidiscrete solution