No abstract

In this paper we study global well-posedness and long time asymptotic behavior of solutions to the nonlinear heat equation with absorption, ut −∆u+|u| α u = 0, where u = u(t, x) ∈ R, (t, x) ∈ (0, ∞) × R N and α > 0. We focus particularly on highly singular initial values which are antisymmetric with respect to the variables x1, x2, · · · , xm for some m ∈ {1, 2, · · · , N }, such asIn fact, we show global well-posedness for initial data bounded in an appropriate sense by u0, for any α > 0.Our approach is to study well-posedness and large time behavior on sectorial domains of the form Ωm = {x ∈ R N : x1, · · · , xm > 0}, and then to extend the results by reflection to solutions on R N which are antisymmetric. We show that the large time behavior depends on the relationship between α and 2/(γ + m), and we consider all three cases, α equal to, greater than, and less than 2/(γ + m). Our results include, among others, new examples of self-similar and asymptotically self-similar solutions.

No abstract