Existence and asymptotic behaviour of solutions for a viscous p(x)-Laplacian equation

Abstract

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 0 in -norm as t → ∞ and under some additional conditions, the solution approaches its initial data in L ²(0, T; H ¹(Ω))-norm as p ⁻ → ∞ where .

Keywords:

• nonlinear parabolic equations
• nonstandard growth conditions
• p(x)-Laplacian operator
• Asymptotic behaviour

AMS Subject Classifications::

• 35K55
• 35K65
• 35D10

Acknowledgements

This research was supported by NSFC (10771085) in the beginning Key Lab of Symbolic Computation and Knowledge Engineering of Ministry of Education and the 985 program of Jilin University. The authors thank the referees for their helpful comments and suggestions.