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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
##### Keywords:
similarity variables
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