Periodic solutions of non-Newtonian polytropic filtration equations with nonlinear sources

Abstract

In this paper, the authors first consider the Dirichlet boundary value problem to the non-Newtonian polytropic filtration equation of the form ∂ u ∂ t = div ( | ∇ u m | p - 2 ∇ u m ) + h ( x , t ) u α , in Ω × R with strong nonlinear sources. The existence of nontrivial periodic solutions is established based on topological degree theory. The authors also studied the Dirichlet boundary value problem to the equation in the form ∂ u ∂ t = div | ∇ ( | u | m - 1 u ) | p - 2 ∇ ( | u | m - 1 u ) + B ( x , t , u ) + f ( x , t ) , in Ω × R with weak nonlinear sources. The existence is treated with Leray-Schauder fixed point theory.