Advances in Differential Equations

Asymptotic behaviour of solutions of a quasilinear parabolic equation with Robin boundary condition

Michèle Grillot and Philippe Grillot

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Abstract

In this paper we study solutions of the quasi-linear parabolic equations $\partial u/\partial t -{\Delta} _p u = a(x) |u|^{q-1}u$ in $(0,T) \times {\Omega} $ with Robin boundary condition ${\partial} u /{\partial} \nu|\nabla u|^{p-2} = b(x) |u|^{r-1}u$ in $(0,T) \times {\partial} {\Omega}$ where $\Omega$ is a regular bounded domain in ${\mathbb R}^N$, $N \geq 3$, $q>1$, $r>1$ and $p \geq 2$. Some sufficient conditions on $a$ and $b$ are obtained for those solutions to be bounded or blowing up at a finite time. Next we give the asymptotic behavior of the solution in special cases.

Article information

Source
Adv. Differential Equations Volume 17, Number 5/6 (2012), 401-419.

Dates
First available in Project Euclid: 17 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.ade/1355703075

Mathematical Reviews number (MathSciNet)
MR2951936

Zentralblatt MATH identifier
1257.35042

Subjects
Primary: 35B05: Oscillation, zeros of solutions, mean value theorems, etc. 35B40: Asymptotic behavior of solutions 35B45: A priori estimates 35B50: Maximum principles 35D05 35H30: Quasi-elliptic equations 35K55: Nonlinear parabolic equations

Citation

Grillot, Michèle; Grillot, Philippe. Asymptotic behaviour of solutions of a quasilinear parabolic equation with Robin boundary condition. Adv. Differential Equations 17 (2012), no. 5/6, 401--419. https://projecteuclid.org/euclid.ade/1355703075.


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