Related papers: Abstract Parabolic Equations with boundary white n…

We study a two-point free boundary problem in a sector for a quasilinear parabolic equation. The boundary conditions are assumed to be spatially and temporally "self-similar" in a special way. We prove the existence, uniqueness and…

When the evolution familiy is hyperbolic and satisfies the Acquistapace-Terreni conditions, the existence and uniquenness of an almost automorphic mild solution and a weighted pseudo almost automorphic mild solution in distribution of…

We survey some of our recent results on existence, uniqueness and regularity of function solutions to parabolic and transport type partial differential equations driven by non-differentiable noises. When applied pathwise to random…

The aim article is to contribute to the definition of a versatile language for metastability in the context of partial differential equations of evolutive type. A general framework suited for parabolic equations in one dimensional bounded…

We study the effect of Gaussian perturbations on a class of model hyperbolic partial differential equations with double symplectic characteristics in low spatial dimensions, extending some recent work in [5]. The coefficients of our partial…

We consider stochastic reaction-diffusion equations with colored noise and prove Schauder type estimates, which will depend on the color of the noise, for the stationary and evolution problems associated with the corresponding transition…

We prove a new linearization principle for the nonlinear stability of solutions to semilinear evolution equations of parabolic type. We assume that the set of equilibria forms a finite dimensional manifold of normally stable and normally…

In this paper, we establish the existence and uniqueness of invariant measures for a class of semilinear stochastic partial differential equations driven by multiplicative noise on a bounded domain. The main results can be applied to SPDEs…

This course introduces the use of semigroup methods in the solution of linear and nonlinear (quasi-linear) hyperbolic partial differential equations, with particular application to wave equations and Hermitian hyperbolic systems. Throughout…

We prove the unique solvability in weighted Sobolev spaces of non-divergence form elliptic and parabolic equations on a half space with the homogeneous Neumann boundary condition. All the leading coefficients are assumed to be only…

This paper is interested in semilinear stochastic equations having unbounded nonlinear perturbations in the deterministic part and/or in the random part. Moreover, the linear part of these equations is governed by a not necessarily analytic…

We consider the Cauchy problem for a semilinear stochastic differential inclusion in a Hilbert space. The linear operator generates a strongly continuous semigroup and the nonlinear term is multivalued and satisfies a condition which is…

We study the approximation of SPDEs on the whole real line near a change of stability via modulation or amplitude equations, which acts as a replacement for the lack of random invariant manifolds on extended domains. Due to the…

An initial boundary value problem of the nonlinear diffusion equation with a dynamic boundary condition is treated. The existence problem of the initial-boundary value problem is discussed. The main idea of the proof is an abstract approach…

We consider infinite-dimensional parabolic rough evolution equations. Using regularizing properties of analytic semigroups we prove global-in-time existence of solutions and investigate random dynamical systems for such equations.

These notes are based on a series of lectures given first at the University of Warwick in spring 2008 and then at the Courant Institute, Imperial College London, and EPFL. It is an attempt to give a reasonably self-contained presentation of…

We propose and analyse a new type of fully discrete surface finite element approximation of a class of linear parabolic stochastic evolution equations with additive noise. Our discretization uses a surface finite element approximation of…

The paper examines the existence of pseudo almost periodic solutions to some classes of partial hyperbolic evolution equations. Namely, sufficient conditions for the existence and uniqueness of pseudo almost periodic solutions to those…

We consider a quasi-linear parabolic equation with nonlinear dynamic boundary conditions occurring as a natural generalization of the semilinear reaction-diffusion equation with dynamic boundary conditions. The corresponding class of…

For semilinear stochastic evolution equations whose coefficients are more general than the classical global Lipschitz, we present results on the strong convergence rates of numerical discretizations. The proof of them provides a new…