Density estimates on a parabolic SPDE

Well-posedness and Regularity for Quasilinear Degenerate Parabolic-hyperbolic Spde

We study quasilinear degenerate parabolic-hyperbolic stochastic partial differential equations with general multiplicative noise within the framework of kinetic solutions. Our results are twofold: First, we establish new regularity results based on averaging techniques. Second, we prove the existence and uniqueness of solutions in a full L1 setting requiring no growth assumptions on the nonline...

A Quasilinear Parabolic Equation With Inhomogeneous Density and Absorption

Relating the almost-sure Lyapunov exponent of a parabolic SPDE and its coefficients’ spatial regularity

We derive a lower bound on the large-time exponential behavior of the solution to a stochastic parabolic partial differential equation on R+ × R in the case of a spatially homogeneous Gaussian potential that is white-noise in time, and study the relation between the lower bound and the potential’s spatial modulus of continuity.

Gradient Estimates for a Nonlinear Parabolic Equation on Riemannian Manifolds

Let (M, g) be a complete noncompact Riemannian manifold. In this paper, we derive a local gradient estimate for positive solutions to a simple nonlinear parabolic equation ∂u ∂t = ∆u+ au log u+ bu on M × [0,+∞), where a, b are two real constants. This equation is closely related to the gradient Ricci soliton. We extend the result of L. Ma (Journal of Functional Analysis 241 (2006) 374-382).

On Estimates for Fully Nonlinear Parabolic Equations on Riemannian Manifolds

In this paper we present some new ideas to derive a priori second order estiamtes for a wide class of fully nonlinear parabolic equations. Our methods, which produce new existence results for the initial-boundary value problems in R n , are powerful enough to work in general Riemannian manifolds. Mathematical Subject Classification (2010): 35K10, 35K55, 58J35, 35B45.