Reflections on Dubinskiĭ's Nonlinear Compact Embedding Theorem

John W. Barrett; Endre Süli

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On initial boundary value problems with equivalued surface for nonlinear parabolic equations.

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Existence of solutions for a class of second-order $p$ -Laplacian systems with impulsive effects

Peng Chen, Xianhua Tang (2014)

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The purpose of this paper is to study the existence and multiplicity of a periodic solution for the non-autonomous second-order system $\begin{matrix} \frac{d}{d t} (| \dot{u} (t) |^{p - 2} \dot{u} (t)) = \nabla F (t, u (t)), a.e. t \in [0, T], \\ u (0) - u (T) = \dot{u} (0) - \dot{u} (T) = 0, \\ Δ {\dot{u}}^{i} (t_{j}) = {\dot{u}}^{i} (t_{j}^{+}) - {\dot{u}}^{i} (t_{j}^{-}) = I_{i j} (u^{i} (t_{j})), i = 1, 2, \dots, N; j = 1, 2, \dots, m . \end{matrix}$ By using the least action principle and the saddle point theorem, some new existence theorems are obtained for second-order $p$ -Laplacian systems with or without impulse under weak sublinear growth conditions, improving some existing results in the literature.

Existence of weak solutions to doubly degenerate diffusion equations

Aleš Matas, Jochen Merker (2012)

Applications of Mathematics

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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...

Strongly nonlinear parabolic equations with natural growth terms and $L^{1}$ data in Orlicz spaces.

Elmahi, A., Meskine, D. (2005)

Portugaliae Mathematica. Nova Série

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