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The blow-up rate for a semilinear parabolic equation with a nonlinear boundary condition. (English) Zbl 0924.35017
Summary: In this paper we obtain the blow up for positive solutions of \(u_t=u_{xx}-\lambda u^p\), in \((0,1)\times(0,T)\) with boundary conditions \(u_x(1,t)=u^q(1,t),u_x(0,t)=0\). If \(p<2q-1\) or \(p=2q-1,0<\lambda<q\), we find that the behaviour of \(u\) is given by \(u(1,t)\sim(T-t)^{-\frac{1}{2(q-1)}}\) and if \(\lambda<0\) and \(p\geq 2q-1\), the blow-up rate is given by \(u(1,t)\sim(T-t)^{-\frac{1}{p-1}}\). We also characterize the blow-up profile similarity variables.

MSC:
35B40 Asymptotic behavior of solutions to PDEs
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
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