Following the lead of [Carrillo, Arch. Ration. Mech. Anal. 147 (1999) 269–361], recently several authors have used Kružkov’s device of “doubling the variables” to prove uniqueness results for entropy solutions of nonlinear degenerate parabolic equations. In all these results, the second order differential operator is not allowed to depend explicitly on the spatial variable, which certainly rest...

Using the maximum principle for semicontinuous functions (Differential Integral Equations 3 (1990), 1001–1014; Bull. Amer. Math. Soc. (N.S) 27 (1992), 1–67), we establish a general ‘‘continuous dependence on the nonlinearities’’ estimate for viscosity solutions of fully nonlinear degenerate parabolic equations with timeand space-dependent nonlinearities. Our result generalizes a result by Souga...

We obtain new oscillation and gradient bounds for the viscosity solutions of fully nonlinear degenerate elliptic equations where the Hamiltonian is a sum of a sublinear and a superlinear part in the sense of Barles and Souganidis (2001). We use these bounds to study the asymptotic behavior of weakly coupled systems of fully nonlinear parabolic equations. Our results apply to some “asymmetric sy...

The behavior of the “minimal branch” is investigated for quasilinear eigenvalue problems involving the p-Laplace operator, considered in a smooth bounded domain of RN , and compactness holds below a critical dimension N #. The nonlinearity f (u) lies in a very general class and the results we present are new even for p = 2. Due to the degeneracy of p-Laplace operator, for p = 2 it is crucial to...

In this work we compare two semidiscrete schemes for the solution of hy-perbolic conservation laws, namely the relaxation [JX95] and the Kurganov Tadmor central scheme [KT00]. We are particularly interested in their behavior under small time steps, in view of future applications to convection diffusion problems. The schemes are tested on two benchmark problems, with one space variable. 1 Motiva...

We investigate degenerate cross-diffusion equations, with a rank-deficient diffusion-matrix, modelling multispecies population dynamics driven by partial pressure gradients. These equations have recently been found to arise in mean-field limit of interacting stochastic particle systems. To date, their analysis multiple space dimensions has confined the purely convective case equal mobility coef...

This paper is mainly concerned with the blow-up and global existence profile for the Cauchy problem of a class of fully nonlinear degenerate parabolic equations with reaction sources. MSC: 35B33, 35B40, 35K65, 35K55

We give sufficient conditions under which solutions of discretized in space second-order parabolic and elliptic equations, perhaps degenerate, admit estimates of the first derivatives in the space variables independent of the mesh size.