×

Asymptotic behaviour of solutions of a quasilinear parabolic equation with Robin boundary condition. (English) Zbl 1257.35042

Summary: In this paper we study solutions of the quasilinear parabolic equations \(\partial u / \partial t - \Delta_p u = a(x)|u|^{q-1} u \;\text{in} \;(0, T) \times \Omega\) with Robin boundary condition \(\partial u / \partial v | \nabla u |^{p-2} = b(x)|u|^{r-1} u \;\text{in} \;(0, T ) \times \partial \Omega\), where \(\Omega\) is a regular bounded domain in \( \mathbb {R}^N , N \geq 3, q \geq 1, r \geq 1\) 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.

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35K59 Quasilinear parabolic equations
35K20 Initial-boundary value problems for second-order parabolic equations
35K92 Quasilinear parabolic equations with \(p\)-Laplacian
35K61 Nonlinear initial, boundary and initial-boundary value problems for nonlinear parabolic equations
PDFBibTeX XMLCite