Hyperbolic Limit of Parabolic Semilinear Heat Equations with Fading Memory

On Weak Solutions of Semilinear Hyperbolic-parabolic Equations

In this paper we prove the existence and uniqueness of weak solutions of the mixed problem for the nonlinear hyperbolic-parabolic equation (K (x, t)u’)’ + K(x, t)u’ + A(t)u + F(u) f with null Dirichlet boundary conditions and zero initial data, where F(s) is a continuous function such that sF(s) >_ O, Vs E R and {A(t);t >_ 0} is a family of operators of L(H(2);H-I(gt)) For the existence we appl...

On Multistep-galerkin Discretizations of Semilinear Hyperbolic and Parabolic Equations*

Henceforth, it will be assumed that (1.1) has a unique solution U. Precise hypotheses on the required smoothness of the solution will be made when needed. For s a nonnegative integer, let H” = H”(Q) be the Sobolev space W,“(R) of real-valued functions on R and let 11. /Is denote its usual norm. Let fi’ = ii’(Q) = {u E H’ :ulm = 0). Th e inner product on L* = L’(Q) is denoted by (. , .) and the ...

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 ...

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