# American Institute of Mathematical Sciences

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January  2007, 18(1): 71-84. doi: 10.3934/dcds.2007.18.71

## Blow-up behavior for a quasilinear parabolic equation with nonlinear boundary condition

 1 Department of Mathematics, National Taiwan Normal University, 88, S-4 Ting Chou Road, Taipei 116, Taiwan

Received  May 2006 Revised  November 2006 Published  February 2007

In this paper, we study the solution of an initial boundary value problem for a quasilinear parabolic equation with a nonlinear boundary condition. We first show that any positive solution blows up in finite time. For a monotone solution, we have either the single blow-up point on the boundary or blow-up on the whole domain, depending on the parameter range. Then, in the single blow-up point case, the existence of a unique self-similar profile is proven. Moreover, by constructing a Lyapunov function, we prove the convergence of the solution to the unique self-similar solution as $t$ approaches the blow-up time.
Citation: Jong-Shenq Guo. Blow-up behavior for a quasilinear parabolic equation with nonlinear boundary condition. Discrete & Continuous Dynamical Systems - A, 2007, 18 (1) : 71-84. doi: 10.3934/dcds.2007.18.71
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