The generic gradient-like structure of certain asymptotically autonomous semilinear parabolic equations

On asymptotically symmetric parabolic equations

We consider global bounded solutions of fully nonlinear parabolic equations on bounded reflectionally symmetric domains, under nonhomogeneous Dirichlet boundary condition. We assume that, as t→∞, the equation is asymptotically symmetric, the boundary condition is asymptotically homogeneous, and the solution is asymptotically strictly positive in the sense that all its limit profiles are strictl...

Optimal Control of Nonsmooth, Semilinear Parabolic Equations

This paper is concerned with an optimal control problem governed by a semilinear, nonsmooth operator differential equation. The nonlinearity is locally Lipschitz-continuous and directionally differentiable, but not Gâteaux-differentiable. Two types of necessary optimality conditions are derived, the first one by means of regularization, the second one by using the directional differentiability ...

Heteroclinic orbits of semilinear parabolic equations

Determining nodes for semilinear parabolic equations

We discuss the uniqueness of the equilibria of time-global solutions of general semilinear parabolic equations by a finite set of values of these solutions. More precisely, if the asymptotic behaviour of a time-global solution is known on an appropriate finite set, then the asymptotic behaviour of a time-global solution itself is entirely determined in a domain.

Semilinear Parabolic Equations with Prescribed Energy

In this paper we study the following reaction-di usion equation ut = u+ f(u; k(t)) subject to appropriate initial and boundary conditions, where f(u; k(t)) = u k(t) or k(t)u with p > 1 and k(t) is an unknown function. An additional energy type condition is imposed in order to nd the solution u(x; t) and k(t). This type of problem is frequently encountered in nuclear reaction process, where the ...