A Class of Semilinear Parabolic Problems and Analytic Semigroups

Semilinear parabolic problems

Numerical Analysis of Semilinear Parabolic Problems

In these lectures I discuss error analysis techniques for nite element methods for systems of reaction-diiusion equations with applications in dynamical systems theory. The emphasis is on pedagogical aspects and analysis techniques rather than on results. The list of techniques discussed include: analytic semigroup, parabolic smoothing, non-smooth data error estimate, a priori error estimate, a...

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...

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...

Hybrid Optimal Control Problems for a Class of Semilinear Parabolic Equations

A class of optimal control problems of hybrid nature governed by semilinear parabolic equations is considered. These problems involve the optimization of switching times at which the dynamics, the integral cost, and the bounds on the control may change. Firstand secondorder optimality conditions are derived. The analysis is based on a reformulation involving a judiciously chosen transformation ...