Time-dependent singularities in semilinear parabolic equations: Behavior at the singularities

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• 作者：Toru Kan ; ^{kan@math.titech.ac.jp" class="auth_mail" title="E-mail the corresponding author} ; Jin Takahashi ^{takahashi.j.ab@m.titech.ac.jp" class="auth_mail" title="E-mail the corresponding author}
• 关键词：35K58 ; 35A20 ; 35B60
• 刊名：Journal of Differential Equations
• 出版年：2016
• 出版时间：15 May 2016
• 年：2016
• 卷：260
• 期：10
• 页码：7278-7319
• 全文大小：482 K

文摘

Singularities of solutions of semilinear parabolic equations are discussed. A typical equation is ∂_tu−Δu=u^p${\partial }_{t}u-\mathrm{\Delta }u=u^p$, a246299ea83d5" title="Click to view the MathML source">x∈R^N∖{ξ(t)}$x\in {\mathbb{R}}^N\setminus \{\xi (t)\}$, t∈I$t\in I$. Here N≥2$N\ge 2$, p>1$p>1$, I⊂R$I\subset \mathbb{R}$ is an open interval and ξ∈C^α(I;R^N)$\xi \in C^{\alpha }(I;{\mathbb{R}}^N)$ with α>1/2$\alpha >1/2$. For this equation it is shown that every nonnegative solution u   satisfies ∂_tu−Δu=u^p+Λ${\partial }_{t}u-\mathrm{\Delta }u=u^p+\mathrm{\Lambda }$ in D^′(R^N×I)${\mathcal{D}}^{\prime }({\mathbb{R}}^N\times I)$ for some measure Λ whose support is contained in {(ξ(t),t);t∈I}$\{(\xi (t),t);t\in I\}$. Moreover, if (N−2)p<N$(N-2)p<N$, then u(x,t)=(a(t)+o(1))Ψ(x−ξ(t))$u(x,t)=(a(t)+o(1))\mathrm{\Psi }(x-\xi (t))$ for almost every t∈I$t\in I$ as x→ξ(t)$x\to \xi (t)$, where Ψ is the fundamental solution of Laplace's equation in R^N${\mathbb{R}}^N$ and a is some function determined by Λ.