Heteroclinic Orbits of Semilinear Parabolic Equations

Heteroclinic orbits of semilinear parabolic equations

Eternal Solutions and Heteroclinic Orbits of a Semilinear Parabolic Equation

This dissertation describes the space of heteroclinic orbits for a class of semilinear parabolic equations, focusing primarily on the case where the nonlinearity is a second degree polynomial with variable coefficients. Along the way, a new and elementary proof of existence and uniqueness of solutions is given. Heteroclinic orbits are shown to be characterized by a particular functional being f...

On Connecting Orbits of Semilinear Parabolic Equations on S

It is well-known that any bounded orbit of semilinear parabolic equations of the form ut = uxx + f(u, ux), x ∈ S 1 = R/Z, t > 0, converges to steady states or rotating waves (non-constant solutions of the form U(x − ct)) under suitable conditions on f . Let S be the set of steady states and rotating waves (up to shift). Introducing new concepts — the clusters and the structure of S —, we clarif...

Heteroclinic Orbits for Retarded Functional Differential Equations*

Suppose r is a heteroclinic orbit of a Ck functional differential equation i(t) =f(x,) with a-limit set a(T) and o-limit set w(T) being either hyperbolic equilibrium points or periodic orbits. Necessary and sufficient conditions are given for the existence of an 7 close to f in Ck with the property that i(t) = 3(x,) has a heteroclinic orbit p close to f. The orbits p are obtained from the zeros...

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 ...