The blow-up properties of solutions to semilinear heat equations with nonlinear boundary conditions
The authors consider the problem \(u_t=u_{xx}+u^p\) in \((0,1)\times(0,T)\), \(u_x(0,t)=0\), \(u_x(1,t)=u^q(1,t)\), \(t\in(0,T)\), \(u(x,0)=u_0(x)\geq 0\), \(x\in[0,1]\), where \(p,q>0\) and \(u_0\not\equiv 0\). If \(\max(p,q)>1\) then the solution of this problem blows up in a finite time \(T\) in the \(L^\infty\)-norm. The authors study the corresponding blow-up rate and blow-up set. Denote \(U(t)=\sup_x| u(x,t)| \) and \(\alpha=1/\max(p-1,2q-2)\). Under some additional assumptions on \(u_0\) it is shown that \(0<c\leq U(t)(T-t)^\alpha\leq C<\infty\) and that blow-up occurs only at \(x=1\).
scientific article; zbMATH DE number 1320486