Geodesic
THE BLOW-UP RATE FOR A SEMILINEAR PARABOLIC EQUATION WITH A NONLINEAR BOUNDARY CONDITION
Acta mathematica Universitatis Comenianae, Tome 67 (1998) no. 2 
J. D. Rossi. THE BLOW-UP RATE FOR A SEMILINEAR PARABOLIC EQUATION WITH A NONLINEAR BOUNDARY CONDITION. Acta mathematica Universitatis Comenianae, Tome 67 (1998) no. 2. http://geodesic.mathdoc.fr/item/AMUC_1998_67_2_a7/
@article{AMUC_1998_67_2_a7,
     author = {J. D. Rossi},
     title = {THE {BLOW-UP} {RATE} {FOR} {A} {SEMILINEAR} {PARABOLIC} {EQUATION} {WITH} {A} {NONLINEAR} {BOUNDARY} {CONDITION}},
     journal = {Acta mathematica Universitatis Comenianae},
     year = {1998},
     volume = {67},
     number = {2},
     url = {http://geodesic.mathdoc.fr/item/AMUC_1998_67_2_a7/}
}
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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, 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.