Location of blow-up set for a semilinear parabolic equation with large diffusion
The paper deals with the problem \[ \begin{cases} u_t=D\Delta u+u^p &\text{in}\;\Omega\times(0,T_D),\\ \displaystyle {{\partial u}\over {\partial \nu}}(x,t)=0 &\text{on}\;\partial\Omega\times(0,T_D),\\ u(x,0)=\phi(x)\geq0 &\text{in}\;\Omega, \end{cases} \] where \(\Omega\) is a bounded and smooth domain in \({\mathbb R}^N,\) \(D>0,\) \(T_D>0\) and \(p>1\) with \((N-2)p\leq N+2.\) Set \(P_2\) for the projection of \(L^2(\Omega)\) onto the second Neumann eigenspace. The authors prove that, if \(P_2\phi\not\equiv0\) in \(\Omega\) and if \(D\) is large enough then the solution \(u(x,t)\) of the above problem blows-up only near the set \({\mathcal M}\cup \partial \Omega\), where \({\mathcal M}=\{x\in\overline\Omega: (P_2\phi)(x)=\max_{y\in\overline\Omega} (P_2\phi)(y)\}.\)
scientific article; zbMATH DE number 2078341