A High-order Exponential Integrator for Nonlinear Parabolic Equations with Nonsmooth Initial Data

A High Order Finite Dierence Method for Random Parabolic Partial Dierential Equations

In this paper, for the numerical approximation of random partial differential equations (RPDEs) of parabolic type, an explicit higher order finite difference scheme is constructed. In continuation the main properties of deterministic difference schemes, i.e. consistency, stability and convergency are developed for the random cases. It is shown that the proposed random difference scheme has thes...

On the Discretization in Time of Semilinear Parabolic Equations with Nonsmooth Initial Data

Single-step discretization methods are considered for equations of the form u, + Au = f(t, u), where A is a linear positive definite operator in a Hubert space H. It is shown that if the method is consistent with the differential equation then the convergence is essentially of first order in the stepsize, even if the initial data v are only in H, but also that, in contrast to the situation in t...

The exponential Rosenbrock-Euler method for nonsmooth initial data

We consider the exponential Rosenbrock-Euler method for the solution of nonlinear parabolic abstract ordinary differential equations. For smooth solutions the convergence analysis is known, and the method is of order two. Here we investigate the convergence of the method in the case of nonsmooth initial conditions which result in derivatives of the solution with singularities at the origin. We ...

Attractors for Parabolic Equations with Nonlinear Boundary Conditions, Critical Exponents, and Singular Initial Data

Carleman Estimates and Unique Continuation for Second Order Parabolic Equations with Nonsmooth Coefficients

In this work we obtain strong unique continuation results for variable coefficient second order parabolic equations. The coefficients in the principal part are assumed to satisfy a Lipschitz condition in x and a Hölder C 1 3 condition in time. The coefficients in the lower order terms, i.e. the potential and the gradient potential, are allowed to be unbounded and required only to satisfy mixed ...