Existence of solutions for anisotropic parabolic Ni-Serrin type equations originated from a capillary phenomena with nonstandard growth nonlinearity

Abstract

We consider an initial boundary value problem for a class of anisotropic parabolic Ni-Serrin type equations with nonstandard nonlinearity in a bounded smooth domain with homogeneous Dirichlet boundary condition. Because the nonlinear perturbation leads to difficulties (it does not have a definite sign) in obtaining a priori estimates in the energy method, we had to modify the Tartar method significantly. Under suitable assumptions, we obtain the global existence, decay and extinction of solutions.

Keywords:

• Anisotropic parabolic Ni-Serrin type equations
• variable exponents
• global existence
• extinction of solutions
• decay estimates

AMS SUBJECT CLASSIFICATIONS:

• 35K55
• 35B09
• 35B44

Acknowledgments

The authors would like to thank the referees for the valuable comments and helpful suggestions on the quality improving of our paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).