Large Time Behavior of Solutions to the Nonlinear Heat Equation with Absorption with Highly Singular Antisymmetric Initial Values

Abstract

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 - Δ ⁢ u + | u | α ⁢ u = 0 {u_{t}-\Delta u+\lvert u\rvert^{\alpha}u=0} , where u = u ⁢ ( t , x ) ∈ ℝ {u=u(t,x)\in\mathbb{R}} , ( t , x ) ∈ ( 0 , ∞ ) × ℝ N {(t,x)\in(0,\infty)\times\mathbb{R}^{N}} and α > 0 {\alpha>0} . We focus particularly on highly singular initial values which are antisymmetric with respect to the variables x 1 , x 2 , … , x m {x_{1},x_{2},\ldots,x_{m}} for some m ∈ { 1 , 2 , … , N } {m\in\{1,2,\ldots,N\}} , such as u 0 = ( - 1 ) m ∂ 1 ∂ 2 ⋯ ∂ m | ⋅ | - γ ∈ 𝒮 ′ ( ℝ N ) {u_{0}=(-1)^{m}\partial_{1}\partial_{2}\cdots\partial_{m}\lvert\,{\cdot}\,% \rvert^{-\gamma}\in\mathcal{S}^{\prime}(\mathbb{R}^{N})} , 0 < γ < N {0<\gamma<N} . In fact, we show global well-posedness for initial data bounded in an appropriate sense by u 0 {u_{0}} for any α > 0 {\alpha>0} . Our approach is to study well-posedness and large time behavior on sectorial domains of the form Ω m = { x ∈ ℝ N : x 1 , … , x m > 0 } {\Omega_{m}=\{x\in\mathbb{R}^{N}:x_{1},\ldots,x_{m}>0\}} , and then to extend the results by reflection to solutions on ℝ N {\mathbb{R}^{N}} which are antisymmetric. We show that the large time behavior depends on the relationship between α and 2 γ + m {\frac{2}{\gamma+m}} , and we consider all three cases, α equal to, greater than, and less than 2 γ + m {\frac{2}{\gamma+m}} . Our results include, among others, new examples of self-similar and asymptotically self-similar solutions.