Existence and asymptotic behaviour of solutions for a viscous \(p(x)\)-Laplacian equation. (English) Zbl 1248.35112
Summary: In this article we investigate the existence of weak solutions by constructing a sequence of weak solutions with the use of difference and variation techniques and passing a limit process with some necessary a priori estimates. Also, two types of asymptotic behaviours of the weak solutions are studied. We prove that the solution approaches zero in \(H^1_0(\Omega)\)-norm as \(t \to \infty\) and under some additional conditions, the solution approaches its initial data in \(L^2(0,T;H^1_0(\Omega ))\)-norm as \(p^- \to \infty\) where \(p^- =\mathrm{inf}_\Omega p(x)\).
MSC:
| 35K92 | Quasilinear parabolic equations with \(p\)-Laplacian |
| 35K55 | Nonlinear parabolic equations |
| 35K65 | Degenerate parabolic equations |
| 35D30 | Weak solutions to PDEs |
| 35B45 | A priori estimates in context of PDEs |
Keywords:
difference and variation techniquesReferences:
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