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Existence of weak solutions to doubly degenerate diffusion equations. (English) Zbl 1249.35194
Summary: s: We prove existence of weak solutions to doubly degenerate diffusion equations \(\dot {u}=\Delta _pu^{m-1}+f\) (\(m,p\geq 2\)) by Faedo-Galerkin approximation for general domains and general nonlinearities. More precisely, we discuss the equation in an abstract setting, which allows to choose function spaces corresponding to bounded or unbounded domains \(\Omega \subset \mathbb {R}^n\) with Dirichlet or Neumann boundary conditions. The function \(f\) can be an inhomogeneity or a nonlinearity involving terms of the form \(f(u)\) or \(\text{div}(F(u))\). In the appendix, an introduction to weak differentiability of functions with values in a Banach space appropriate for doubly nonlinear evolution equations is given.

MSC:
35K92 Quasilinear parabolic equations with \(p\)-Laplacian
35D30 Weak solutions to PDEs
37L65 Special approximation methods (nonlinear Galerkin, etc.) for infinite-dimensional dissipative dynamical systems
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