Viscosity solutions to parabolic complex Hessian type equations

Viscosity solutions for quasilinear degenerate parabolic equations of porous medium type

We consider the Cauchy Problem for the class of nonlinear parabolic equations of the form ut = a(u)∆u+ |∇u|, with a function a(u) that vanishes at u = 0. Because of the degenerate character of the coefficient a the usual concept of viscosity solution in the sense of Crandall-Evans-Lions has to be modified to include the behaviour at the free boundary. We prove that the problem is well-posed in ...

Viscosity solutions for elliptic-parabolic problems

We study an elliptic-parabolic problem appearing in the theory of partially saturated flows in the framework of viscosity solutions. This is part of current investigation to understand the theory of viscosity solutions for PDE problems involving free boundaries. We prove that the problem is well posed in the viscosity setting and compare the results with the weak theory. Dirichlet or Neumann bo...

Existence of viscosity multi-valued solutions with asymptotic behavior for Hessian equations

A Generalized Osgood Condition for Viscosity Solutions to Fully Nonlinear Parabolic Degenerate Equations

Using a generalized assumption of Osgood type, we prove a new comparison result for viscosity sub and supersolutions of fully nonlinear, possibly degenerate, parabolic equations. Our result allows to deal with hamiltonian functions with a quadratic growth in the spatial gradient, under special compatibility conditions with the diffusive terms. It applies in particular to a financial differentia...

REGULARITY FOR ENTROPY SOLUTIONS OF PARABOLIC p-LAPLACIAN TYPE EQUATIONS

In this note we give some summability results for entropy solutions of the nonlinear parabolic equation ut − div ap(x,∇u) = f in ]0, T [×Ω with initial datum in L1(Ω) and assuming Dirichlet’s boundary condition, where ap(., .) is a Carathéodory function satisfying the classical Leray-Lions hypotheses, f ∈ L1(]0, T [×Ω) and Ω is a domain in RN . We find spaces of type Lr(0, T ;Mq(Ω)) containing ...