Enrique Zuazua

Null and approximate controllability for weakly blowing up semilinear heat equations

Abstract We consider the semilinear heat equation in a bounded domain of R d , with control on a subdomain and homogeneous Dirichlet boundary conditions. We prove that the system is null-controllable at any time provided a globally defined and bounded trajectory exists and the nonlinear term f(y) is such that |f(s)| grows slower than |s|log3/2(1+|s|) as |s|→∞ . For instance, this condition is fulfilled by any function f growing at infinity like |s|logp(1+|s|) with 1 2 , null controllability does not hold. The problem remains open when f behaves at infinity like |s|logp(1+|s|) , with 3/2≤p≤2 . Results of the same kind are proved in the context of approximate controllability.

Decay estimates for some semilinear damped hyperbolic problems

AbstractLet Ω be a bounded open domain in Rn, g ∶ R → R a non-decreasing continuous function such that g(0)=0 and h ε Lloc1(R+; L2(Ω)). Under suitable assumptions on g and h, the rate of decay of the difference of two solutions is studied for some abstract evolution equations of the general form u′′ + Lu + g(u′) = h(t,x) as t → + ∞. The results, obtained by use of differential inequalities, can be applied to the case of the semilinear wave equation

Controllability and Observability of Partial Differential Equations: Some Results and Open Problems

u_u - \Delta u + g{\text{(}}u_t {\text{) = }}h{\text{ in }}R^ + \times \Omega ,{\text{ }}u = {\text{0 on }}R^ + \times \partial \Omega

On the Controllability of Parabolic Systems with a Nonlinear Term Involving the State and the Gradient

in R+×Ω, u=0 onR+×∂Ω. For instance if

DECAY OF SOLUTIONS OF THE SYSTEM OF THERMOELASTICITY OF TYPE III

g(s) = c\left| s \right|^{p - 1} s + d\left| s \right|^{q - 1} s

Finite dimensional null controllability for the semilinear heat equation

with c, d>0 and 1 < p≦q, (n−2)q≦n+2, then if