This paper presents the numerical solution of a constrained optimal control problem (COCP) for quasilinear parabolic equations. The COCP is converted to unconstrained optimization problem (UOCP) by applying the exterior penalty function method. Necessary optimality conditions for the considered problem are established. The computing optimal controls are helped to identify the unknown coefficien...

The purpose of the talk is to give a review on effective methods for the numerical solution of hyperbolic systems with parabolic relaxation. We consider mathematical models described by a set of hyperbolic equations with relaxation. As the relaxation parameter vanishes, the characteristic speeds of the system diverge, and the system reduces to a parabolic-type equation (typically a convection-d...

We present an abstract framework for semilinear parabolic problems based on analytic semigroup theory. The same framework is used for numerical discretization based on the finite element method. We prove local existence of solutions and local error estimates. These are applied in the context of dynamical systems. The framework is also used to analyze the finite element method for a stochastic p...

In this paper, we introduce a probabilistic numerical scheme for a class of parabolic and elliptic fully non-linear PDEs in bounded domains. In the main result, we provide the convergence of a discrete-time approximation to the viscosity solution of a fully non-linear parabolic equation by assuming that comparison principle holds for the PDE.

We observe that the comparison result of Barles-Biton-Ley for viscosity solutions of a class of nonlinear parabolic equations can be applied to a geometric fully nonlinear parabolic equation which arises from the graphic solutions for the Lagrangian mean curvature flow.

The theory of maximal monotone operators is utilized to prove the existence of weak periodic solutions for a coupled nonlinear parabolic-elliptic system.The diffusion term in the parabolic equation contains a Leray-Lions operator.This system may be regarded as a generalized version of the evolution thermistor model.

Existence, uniqueness, and a priori estimates for solutions are studied for stochastic parabolic Ito equations. An analog of the second fundamental inequality and the related existence theorem are obtained for backward stochastic parabolic Ito equation. AMS 1991 subject classification: Primary 60J55, 60J60, 60H10. Secondary 34F05, 34G10.

We study a seawater intrusion problem in a confined aquifer. This process can be formulated as a coupled system of partial differential equations which includes an elliptic and a degenerate parabolic equation. Existence results of weak solutions, under realistic assumptions, are established through time discretization combined with parabolic regularization.

A new class of random partial differential equations of parabolic type is considered where the stochastic term consists in an irregular noisy drift, not necessarily Gaussian, for which a suitable interpretation is provided. After freezing a realization of the drift (stochastic process), we study existence and uniqueness (in some appropriate sense) of the associated parabolic equation and a prob...

We study the existence of optimal controls for problems governed by semilinear parabolic equations. The nonlinearities in the state equation need not be monotone and the data need not be regular. In particular, the control may be any bounded Radon measure. Our examples include problems with nonlinear boundary conditions and parabolic systems.