ACTA MATHEMATICA UNIVERSITATIS COMENIANAE

ACTA MATHEMATICA UNIVERSITATIS COMENIANAE

Vol. 67, 2 (1998)
pp. 343-350

THE BLOW-UP RATE FOR A SEMILINEAR PARABOLIC EQUATION WITH A NONLINEAR BOUNDARY CONDITION
J. D. ROSSI

Abstract. In this paper we obtain the blow-up rate 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)^-\frac12(q-1)$ and, if $\lambda<0$ and $p \ge 2q-1$, the blow up rate is given by $ u(1 ,t) \sim (T-t)^-\frac1p-1$. We also characterize the blow-up profile in similarity variables.

AMS subject classification. 35B40, 35J65, 35K60
Keywords. blow-up, asymptotic behaviour, nonlinear boundary conditions

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Acta Mathematica Universitatis Comenianae
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