Periodic solutions of non-Newtonian polytropic filtration equations with nonlinear sources. (English) Zbl 1425.35092
Summary: The authors first consider the Dirichlet boundary value problem to the non-Newtonian polytropic filtration equation of the form
\[
\frac{\partial u}{\partial t}=\mathrm{div}(|\nabla u^m|^{p-2}\nabla u^m)+h(x,t)u^2, \text{ in } \Omega \times \mathbb R
\]
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
\[
\frac{\partial u}{\partial t} = \mathrm{div} [|\nabla (|u|^{m-1}u)|^{p-2}\nabla (|u|^{m-1}u)]+B(x,t,u) + f(x,t), \text{ in } \Omega \times \mathbb R
\]
with weak nonlinear sources. The existence is treated with Leray-Schauder fixed point theory.
Keywords:
non-Newtonian polytropic filtration; periodic; nonlinear sources; Leray-Schauder’s degree; topological degree theoryReferences:
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