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 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.

中文翻译：

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具有高奇异反对称初值的吸收非线性热方程解的大时间特性

摘要 在本文中，我们研究了具有吸收的非线性热方程解的全局适定性和长时间渐近行为，ut - Δ ⁢ u + | 你| α ⁢ u = 0 {u_{t}-\Delta u+\lvert u\rvert^{\alpha}u=0} ，其中 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}} 和α > 0 {\alpha>0} 。我们特别关注高度奇异的初始值，这些初始值相对于变量 x 1 , x 2 , … , xm {x_{1},x_{2},\ldots,x_{m}} 对于某些 m ∈ { 1 , 2 , … , N } {m\in\{1,2,\ldots,N\}} , 如 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 0 {\alpha>0 } . 我们的方法是研究 Ω m = { x ∈ ℝ N : x 1 , … , xm > 0 } {\Omega_{m}=\{x\in\mathbb {R}^{N}:x_{1},\ldots,x_{m}>0\}} ，然后将结果通过反射扩展到 ℝ N {\mathbb{R}^{N}} 上的解是反对称的。我们表明大时间行为取决于 α 和 2 γ + m {\frac{2}{\gamma+m}} 之间的关系，我们考虑所有三种情况，α 等于、大于和小于 2 γ + m {\frac{2}{\gamma+m}} 。我们的结果包括自相似和渐近自相似解决方案的新例子。然后通过反射将结果扩展到 ℝ N {\mathbb{R}^{N}} 上的反对称解。我们表明大时间行为取决于 α 和 2 γ + m {\frac{2}{\gamma+m}} 之间的关系，我们考虑所有三种情况，α 等于、大于和小于 2 γ + m {\frac{2}{\gamma+m}} 。我们的结果包括自相似和渐近自相似解决方案的新例子。然后通过反射将结果扩展到 ℝ N {\mathbb{R}^{N}} 上的反对称解。我们表明大时间行为取决于 α 和 2 γ + m {\frac{2}{\gamma+m}} 之间的关系，我们考虑所有三种情况，α 等于、大于和小于 2 γ + m {\frac{2}{\gamma+m}} 。我们的结果包括自相似和渐近自相似解决方案的新例子。