Asymptotic behavior of global solutions of the $u_t=\Delta u + u^{p}$

TL;DR

This paper investigates the long-term behavior of solutions to a nonlinear heat equation, establishing decay rates for solutions that exist globally, using entropy methods to analyze their asymptotic properties.

Contribution

It introduces an entropy-based approach to determine decay rates of global solutions for the semilinear heat equation, extending understanding of solution behavior beyond blow-up scenarios.

Findings

Decay rates for global solutions are established.

Solutions decay at a specific rate depending on initial conditions.

Entropy methods effectively analyze asymptotic behavior.

Abstract

We study the asymptotic behavior of nonnegative solutions of the semilinear parabolic problem {u_t=\Delta u + u^{p}, x\in\mathbb{R}^{N}, t>0 u(0)=u_{0}, x\in\mathbb{R}^{N}, t=0. It is known that the nonnegative solution $u(t)$ of this problem blows up in finite time for $1<p\leq 1+ 2/N$. Moreover, if $p> 1+ 2/N$ and the norm of $u_{0}$ is small enough, the problem admits global solution. In this work, we use the entropy method to obtain the decay rate of the global solution $u(t)$.