Large time behavior of solutions to the nonlinear heat equation with absorption with highly singular antisymmetric initial values

TL;DR

This paper investigates the long-term behavior of solutions to a nonlinear heat equation with absorption, focusing on highly singular antisymmetric initial data, and identifies how the asymptotic behavior depends on the parameters involved.

Contribution

It provides the first analysis of global well-posedness and asymptotic behavior for solutions with highly singular antisymmetric initial values in the nonlinear heat equation.

Findings

Established global well-posedness for all positive lpha>0.

Characterized large time asymptotics depending on lpha and initial data parameters.

Constructed new examples of self-similar and asymptotically self-similar solutions.

Abstract

In this paper we study global well-posedness and long time asymptotic behavior of solutions to the nonlinear heat equation with absorption, $ u_t - \Delta u + |u|^\alpha u =0$, where $u=u(t,x)\in {\mathbb R}, $ $(t,x)\in (0,\infty)\times{\mathbb R}^N$ and $\alpha>0$. We focus particularly on highly singular initial values which are antisymmetric with respect to the variables $x_1,\; x_2,\; \cdots,\; x_m$ for some $m\in \{1,2, \cdots, N\}$, such as $u_0 = (-1)^m\partial_1\partial_2 \cdots \partial_m|\cdot|^{-\gamma} \in {{\mathcal S'}({\mathbb R}^N)}$, $0 < \gamma < N$. In fact, we show global well-posedness for initial data bounded in an appropriate sense by $u_0$, for any $\alpha>0$. Our approach is to study well-posedness and large time behavior on sectorial domains of the form $\Omega_m = \{x \in {{\mathbb R}^N} : x_1, \cdots, x_m > 0\}$, and then to extend the results by reflection to solutions on ${{\mathbb R}^N}$ which are antisymmetric. We show that the large time behavior depends on the relationship between $\alpha$ and $2/(\gamma+m)$, and we consider all three cases, $\alpha$ equal to, greater than, and less than $2/(\gamma+m)$. Our results include, among others, new examples of self-similar and asymptotically self-similar solutions.