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Blow-up analysis for a semilinear parabolic equation with nonlinear memory and nonlocal nonlinear boundary condition. (English) Zbl 1207.35076

Summary: We consider a semilinear parabolic equation
\[ u_t=\Delta u+u^q\int_0^tu^p(x,s)\,ds,\quad x\in \Omega,\;t>0 \]
with nonlocal nonlinear boundary condition \(u|_{\partial\Omega\times(0,+\infty)}=\int_\Omega\varphi(x,y) u^l(y,t)\,dy\) and nonnegative initial data, where \(p, q\geq 0\) and \(l>0\). The blow-up criteria and the blow-up rate are obtained.

MSC:

35B44 Blow-up in context of PDEs
35B35 Stability in context of PDEs
35K58 Semilinear parabolic equations
35K61 Nonlinear initial, boundary and initial-boundary value problems for nonlinear parabolic equations
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