Fractional heat equations with subcritical absorption having a measure as initial data

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• 作者：Huyuan Chen^a ; ^{chenhuyuan@yeah.net" class="auth_mail" title="E-mail the corresponding author} ; Laurent Vé ; ron^b ; ^{Laurent.Veron@lmpt.univ-tours.fr" class="auth_mail" title="E-mail the corresponding author} ; Ying Wang^c ; ^{yingwang00@126.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：35R06 ; 35K05 ; 35R11
• 刊名：Nonlinear Analysis
• 出版年：2016
• 出版时间：May 2016
• 年：2016
• 卷：137
• 期：Complete
• 页码：306-337
• 全文大小：764 K

文摘

We study existence and uniqueness of weak solutions to (F) ∂_tu+(−Δ)^αu+h(t,u)=0${\partial }_{t}u+{(-\Delta )}^{\alpha }u+h(t,u)=0$ in (0,∞)×R^N$(0,\infty )\times {\mathbb{R}}^N$, with initial condition u(0,⋅)=ν$u(0,\cdot )=\nu$ in R^N${\mathbb{R}}^N$, where N≥2$N\ge 2$, the operator (−Δ)^α${(-\Delta )}^{\alpha }$ is the fractional Laplacian with α∈(0,1)$\alpha \in (0,1)$, ν$\nu$ is a bounded Radon measure and h:(0,∞)×R→R$h:(0,\infty )\times \mathbb{R}\to \mathbb{R}$ is a continuous function satisfying a subcritical integrability condition.

In particular, if h(t,u)=t^βu^p$h(t,u)=t^{\beta }{u}^p$ with β>−1$\beta >-1$ and , we prove that there exists a unique weak solution u_k$u_k$ to (F) with ν=kδ₀$\nu =k{\delta }_0$, where δ₀${\delta }_0$ is the Dirac mass at the origin. We obtain that u_k→∞$u_k\to \infty$ in (0,∞)×R^N$(0,\infty )\times {\mathbb{R}}^N$ as k→∞$k\to \infty$ for p∈(0,1]$p\in (0,1]$ and the limit of u_k$u_k$ exists as k→∞$k\to \infty$ when $1<p<p_{\beta }^{\ast }$, we denote it by u_∞$u_{\infty }$. When , u_∞$u_{\infty }$ is the minimal self-similar solution of (F)_∞${\left(F\right)}_{\infty }$∂_tu+(−Δ)^αu+t^βu^p=0${\partial }_{t}u+{(-\Delta )}^{\alpha }u+t^{\beta }{u}^p=0$ in (0,∞)×R^N$(0,\infty )\times {\mathbb{R}}^N$ with the initial condition u(0,⋅)=0$u(0,\cdot )=0$ in R^N∖{0}${\mathbb{R}}^N\setminus \left\{0\right\}$ and it satisfies u_∞(0,x)=0$u_{\infty }(0,x)=0$ for x≠0$x\ne 0$. While if $1<p<p_{\beta }^{\ast \ast }$, then u_∞≡U_p$u_{\infty }\equiv U_p$, where U_p$U_p$ is the maximal solution of the differential equation y^′+t^βy^p=0$y^{\prime }+t^{\beta }{y}^p=0$ on R₊${\mathbb{R}}_{+}$.