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

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 $min {1,2, cdots, N}$, such as $u_0 = (-1)^mpartial_1partial_2 cdots partial_m|cdot|^{-gamma} in {{mathcal S}({mathbb R}^N)}$, $0 < gamma < N$. 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.