The balance between diffusion and absorption in semilinear parabolic equations

The balance between diffusion and absorption in semilinear parabolic equations

Let h : [0,∞) 7→ [0,∞) be continuous and nondecreasing, h(t) > 0 if t > 0, and m,q be positive real numbers. We investigate the behavior when k → ∞ of the fundamental solutions u = uk of ∂tu − ∆u m + h(t)u = 0 in Ω × (0, T ) satisfying uk(x, 0) = kδ0. The main question is wether the limit is still a solution of the above equation with an isolated singularity at (0, 0), or a solution of the asso...

Diffusion versus absorption in semilinear parabolic problems

We study the limit, when k → ∞, of the solutions u = uk of (E) ∂tu−∆u+ h(t)uq = 0 in RN × (0,∞), uk(., 0) = kδ0, with q > 1, h(t) > 0. If h(t) = e−ω(t)/t where ω > 0 satisfies to R 1 0 p ω(t)t−1dt < ∞, the limit function u∞ is a solution of (E) with a single singularity at (0, 0), while if ω(t) ≡ 1, u∞ is the maximal solution of (E). We examine similar questions for equations such as ∂tu−∆u + h...

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 ...

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.

Heteroclinic orbits of semilinear parabolic equations