Periodic solutions of singular linear and semilinear parabolic problems

Periodic Solutions and Optimization Problems for a Class of Semilinear Parabolic Control Systems

This paper is devoted to the study of the existence of periodic solutions for a class of control problems described by a semilinear parabolic equation. Related optimization problems are also considered. Periodic control problems and optimal periodic control problems for evolution equations described both by ordinary differential equations and by parabolic equations arise in many different situa...

Semilinear parabolic problems

Explicit multiple singular periodic solutions and singular soliton solutions to KdV equation

 Based on some stationary periodic solutions and stationary soliton solutions, one studies the general solution for the relative lax system, and a number of exact solutions to the Korteweg-de Vries (KdV) equation are first constructed by the known Darboux transformation, these solutions include double and triple singular periodic solutions as well as singular soliton solutions whose amplitude d...

Boundary value problems and periodic solutions for semilinear evolution inclusions

We consider boundary value problems for semilinear evolution inclusions. We establish the existence of extremal solutions. Using that result, we show that the evolution inclusion has periodic extremal trajectories. These results are then applied to closed loop control systems. Finally, an example of a semilinear parabolic distributed parameter control system is worked out in detail.

Homogenization of periodic semilinear parabolic degenerate PDEs

In this paper a second order semilinear parabolic PDE with rapidly oscillating coefficients is homogenized. The novelty of our result lies in the fact that we allow the second order part of the differential operator to be degenerate in some part of Rd . Our fully probabilistic method is based on the deep connection between PDEs and BSDEs and the weak convergence of a class of diffusion processe...