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