An extinction delay mechanism for abstract semilinear equations
by
Alfonso C. Casal
Universidad Politecnica de Madrid, Spain
Coauthors: Jesus Ildefonso Díaz, Jose Manuel Vegas

We consider an abstract semilinear problem (ASP):u′+Au+F(u)=G(t,u(t+s)), s∈[-r,0]), t∈(0,T),and u=u(s), for s∈(-r,0), given as initial datum, where T>0, A:D(A)→ X is a linear m-accretive operator on a Banach space X, F:X→X is a C¹ function and G:[0,T)×C([-r,0]:X)→X is a suitable delayed action.

By using a nonlinear variation of constants formula (the Alekseev formula) we show that given r∈(0,T/2], A and F, and for any initial datum there exists a delayed action G such that the solution of (ASP) becomes u(t)=0 in X for any t≥2r.

We also show that in the linear case (F(u)≡0) the conclusion holds when the delayed action is of the form G=-b(t)u(t-r) for a suitable real function b(t), different from zero only in [r,2r] (typical of switched controls).

The results generalize those in a previous work (to appear in Dynamics of Continuous, Discrete and Impulsive Systems) for linear scalar parabolic equations. By suitable choices of X, A and F, we apply the abstract result to some models such as general linear abstract equations and some nonlinear damped hyperbolic equations, among many other applications.

Date received: March 6, 2007