A generalized Osgood condition for viscosity solutions to fully nonlinear parabolic degenerate equations (original) (raw)

Viscosity solutions for systems of parabolic variational inequalities

Bernoulli, 2010

where ∂ϕ is the subdifferential operator of the proper convex lower semicontinuous function ϕ : R k → (−∞, +∞] and Lt is a second differential operator given by Ltvi(x) = 1 2 Tr[σ(t, x)σ * (t, x)D 2 vi(x)] + b(t, x), ∇vi(x) , i ∈ 1, k. We prove the uniqueness of the viscosity solution and then, via a stochastic approach, prove the existence of a viscosity solution u : [0, T ] × R d → R k of the above parabolic variational inequality.

Large time behavior of solutions for parabolic equations with nonlinear gradient terms

Hokkaido Mathematical Journal, 2007

In this paper we prove the global existence of mild solutions for the semilinear parabolic equation ut = ∆u + a|∇u| q + b|u| p−1 u, t > 0, x ∈ R n , n ≥ 1, a ∈ R, b ∈ R, p > 1 + (2/n), (n + 2)/(n + 1) < q < 2 and q ≤ p(n + 2)/(n + p), with small initial data with respect to a norm related to the equation. We also prove that some of these global solutions are asymptotic to self-similar solutions of the equations ut = ∆u + νa|∇u| q + µb|u| p−1 u, with ν, µ = 0 or 1. The values of ν and µ depend on the decaying of the initial data and on the position of q with respect to 2p/(p + 1). Our results apply for the viscous Hamilton-Jacobi equation: ut = ∆u + a|∇u| q and hold without sign restriction neither on a nor on the initial data. We prove that if (n + 2)/(n + 1) < q < 2 and the initial data behaves, near infinity, like c|x| −α , (2 − q)/(q − 1) ≤ α < n, c is a small constant, then the resulting solution is global. Moreover, if (2 − q)/(q − 1) < α < n, the solution is asymptotic to a self-similar solution of the linear heat equation. Whereas, if (2 − q)/(q − 1) = α < n, the solution is asymptotic to a self-similar solution of the viscous Hamilton-Jacobi equation. The asymptotics are given, in particular, in W 1, ∞ (R n).

Large time behavior for some nonlinear degenerate parabolic equations

Journal de Mathématiques Pures et Appliquées, 2014

We study the asymptotic behavior of Lipschitz continuous solutions of nonlinear degenerate parabolic equations in the periodic setting. Our results apply to a large class of Hamilton-Jacobi-Bellman equations. Defining Σ as the set where the diffusion vanishes, i.e., where the equation is totally degenerate, we obtain the convergence when the equation is uniformly parabolic outside Σ and, on Σ, the Hamiltonian is either strictly convex or satisfies an assumption similar of the one introduced by Barles-Souganidis (2000) for first-order Hamilton-Jacobi equations. This latter assumption allows to deal with equations with nonconvex Hamiltonians. We can also release the uniform parabolic requirement outside Σ. As a consequence, we prove the convergence of some everywhere degenerate second-order equations.

Pointwise behaviour of semicontinuous supersolutions to a quasilinear parabolic equation

Annali di Matematica Pura ed Applicata, 2006

A kind of supersolutions of the so-called p-parabolic equation are studied. These p-superparbolic functions are defined as lower semicontinuous functions obeying the comparison principle. Incidentally, they are precisely the viscosity supersolutions. One of our results guarantees the existence of a spatial Sobolev gradient. For p = 2 we have the supercaloric functions and the ordinary heat equation.