E. Fernandez-cara et E. Zuazua, Null and approximate controllability for weakly blowing up semilinear heatequations, ANN IHP-AN, 17(5), 2000, pp. 583-616
Citations number
19
Categorie Soggetti
Mathematics
Journal title
ANNALES DE L INSTITUT HENRI POINCARE-ANALYSE NON LINEAIRE
We consider the semilinear heat equation in a bounded domain of R-d, with c
ontrol on a subdomain and homogeneous Dirichlet boundary conditions. We pro
ve that the system is null-controllable at any time provided a globally def
ined and bounded trajectory exists and the nonlinear term f(y) is such that
/f(s)/ grows slower than /s/ log(3/2)(1 + /s/) as /s/ --> infinity. For in
stance, this condition is fulfilled by any function f growing at infinity l
ike /s/ log(P)(1 + /s/) with 1 < P < 3/2 (in this case, in the absence of c
ontrol, blow-up occurs). We also prove that, for some functions f that beha
ve at infinite like /s/ log(P)(1 + /s/) with P > 2, null controllability do
es not hold. The problem remains open when f behaves at infinity like /s/ l
og(P)(1 + /s/), with 3/2 less than or equal to p less than or equal to 2, R
esults of the same kind are proved in the context of approximate controllab
ility. (C) 2000 Editions scientifiques et medicales Elsevier SAS. AMS class
ification: 93B05, 93C20.