Abstract. This paper studies the global regularity theory for degenerate nonlinear parabolic partial differential equations. Our objective is to show that weak solutions belong to a higher Sobolev space than assumed a priori if the complement of the domain satisfies a capacity density condition and if the boundary values are sufficiently smooth. Moreover, we derive integrability estimates for t...

We propose a second order finite volume scheme for nonlinear degenerate parabolic equations. For some of these models (porous media equation, drift-diffusion system for semiconductors, ...) it has been proved that the transient solution converges to a steady-state when time goes to infinity. The present scheme preserves steady-states and provides a satisfying long-time behavior. Moreover, it re...

Abstract This paper deals with the blow-up properties of the solution to the degenerate and singular parabolic system with nonlocal sources, absorptions and homogeneous Dirichlet boundary conditions. The existence of a unique classical nonnegative solution is established and the sufficient conditions for the solution to exist globally or blow up in finite time are obtained. Furthermore, under c...

Abstract. On one hand, the existence of a solution to degenerate parabolic equations, without a nonlinear convection term, can be proven using the results of Alt and Luckhaus, Minty and Kolmogorov. On the other hand, the proof of uniqueness of an entropy weak solution to a nonlinear scalar hyperbolic equation, first provided by Krushkov, has been extended in two directions: Carrillo has handled...

We prove higher integrability of the spatial gradient weak solutions to parabolic systems with $\phi$-growth, where $\varphi=\varphi(t)$ is a general Orlicz function. The need be neither degenerate nor singular. Our result generalized version one J. Kinnunen and Lewis [Duke Math. 102 (2000), no. 2, 253--271] for $p$-Laplace systems.

In this paper we present new efficient variants of structured preconditioners for algebraic linear systems arising from the mortar discretization of a degenerate parabolic system of equations. The new approaches extend and adapt the idea of substructuring preconditioners to the discretization of a degenerate problem in electrocardiology. A polylogarithmic bound for the condition number of the p...

Gradient schemes is a framework that enables the unified convergence analysis of many numerical methods for elliptic and parabolic partial differential equations: conforming and non-conforming Finite Element, Mixed Finite Element and Finite Volume methods. We show here that this framework can be applied to a family of degenerate non-linear parabolic equations (which contain in particular the Ri...

Solvability of forward–backward stochastic di erential equations with nonsmooth coe cients is considered using the Four-Step Scheme and some approximation arguments. For the onedimensional case, the existence of an adapted solution is established for the equation which allows the di usion in the forward equation to be degenerate. As a byproduct, we obtain the existence of a viscosity solution t...