Geodesic
Existence of weak solutions to doubly degenerate diffusion equations
Applications of Mathematics, Tome 57 (2012) no. 1, pp. 43-69.

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We prove existence of weak solutions to doubly degenerate diffusion equations \begin {equation*} \dot {u} = \Delta _p u^{m-1} + f \quad (m,p \ge 2) \end {equation*} 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 $\div (F(u))$. In the appendix, an introduction to weak differentiability of functions with values in a Banach space appropriate for doubly nonlinear evolution equations is given.
DOI : 10.1007/s10492-012-0004-0
Classification : 35A01, 35D30, 35K20, 35K59, 35K65, 35K92, 37L65
Keywords: $p$-Laplacian; doubly nonlinear evolution equation; weak solution 
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Matas, Aleš; Merker, Jochen. Existence of weak solutions to doubly degenerate diffusion equations. Applications of Mathematics, Tome 57 (2012) no. 1, pp. 43-69. doi : 10.1007/s10492-012-0004-0. http://geodesic.mathdoc.fr/articles/10.1007/s10492-012-0004-0/

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