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+\lvert u\rvert^{\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},\ldots,x_{m}} for some {m\in\{1,2,\ldots,N\}}, such as {u_{0}=(-1)^{m}\partial_{1}\partial_{2}\cdots\partial_{m}\lvert\,{\cdot}\,% \rvert^{-\gamma}\in\mathcal{S}^{\prime}(\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},\ldots,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 α and {\frac{2}{\gamma+m}}, and we consider all three cases, α equal to, greater than, and less than {\frac{2}{\gamma+m}}. Our results include, among others, new examples of self-similar and asymptotically self-similar solutions.