Reflections on Dubinskiĭ's Nonlinear Compact Embedding Theorem

John W. Barrett; Endre Süli

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We prove existence of weak solutions to doubly degenerate diffusion equations $\dot{u} = Δ_{p} u^{m - 1} + f (m, p \geq 2)$ 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 $Ω \subset ℝ^{n}$ 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...

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