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

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.