REGULARITY OF THE ASYMPTOTIC BLOW-UP FOR A NONLINEAR PARABOLIC EQUATION WITH CRITICAL ENERGY

Abstract

We consider the behaviour of the critical energy for the nonlinear heat equation. We focus on the analysis of critical solutions for a class of nonlinear heat equations in a bounded domain. In order to study the construction of suitable approximate solutions, and then the more precisely, we show that the solution of a semi-discrete form of (P) becomes zero when t goes to infinity and we give to infinity and give its asymptotic behaviour. Using some non-standard schemes, we also prove certain estimates for discrete forms of the problem  that the numerical approximation of the explosion time

$$
(P) \begin{cases}u_t(x, t)-\kappa \Delta u(x, t)+\varepsilon f(u(x, t))=0, & x \in \Omega, t \in(0, T) \\ u(x, t)=0, & x \in \partial \Omega, t \in(0, T) \\ u(x, 0)=u_0(x), & x \in \Omega,\end{cases}
$$

where $f$ is an increasing $C^1$ function, $\kappa \geq 0, \varepsilon \geq 0$. We derive some criteria for the initial data $u_0$ that guarantee the explosion of the solution of $(P)$ in a finite time. More general non-linearities $f(u(t, x))$ will be briefly considered. We will always assume that $u_0$ belongs to a space of functions $X$ in which the problem $(P)$ is well posed and we denote by $T$ the maximum existence time of the solution of $(P)$. We start with a simple criterion. In the case of a bounded domain, it is based on Kaplan's eigenfunction method, see [9]. We show that the solution of a semi-discrete form of $(P)$ becomes zero when $t$ tends to infinity and give its asymptotic behaviour. Using non-standard schemes, we also prove some estimates for discrete forms of $(P)$. Finally, the heat equation in $(P)$ has been solved numerically by testing the convergence and stability of explicit and implicit schemes in simulation using finite difference methods. The examples show that the implemented schemes are consistent with the theoretical predictions and that the truncation errors depend on the mesh size, spacing and time step. Some numerical examples are given to illustrate all these results.

Received: October 22, 2024
Revised: December 1, 2024
Accepted: January 25, 2025