Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-27T22:25:13.889Z Has data issue: false hasContentIssue false
  • English
  • Français

Well posedness and control of semilinear wave equations with iterated logarithms

Published online by Cambridge University Press:  15 August 2002

Piermarco Cannarsa ,
Vilmos Komornik  and
Paola Loreti
Piermarco Cannarsa
Affiliation:
Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica, 00133 Roma, Italy; cannarsa@axp.mat.uniroma2.it.
Vilmos Komornik
Affiliation:
Institut de Recherche Mathématique Avancée, Université Louis Pasteur et CNRS, 7 rue René Descartes, 67084 Strasbourg Cedex, France; komornik@math.u-strasbg.fr.
Paola Loreti
Affiliation:
Istituto per le Applicazioni del Calcolo “Mauro Picone”, Consiglio Nazionale delle Ricerche, Viale del Policlinico 137, 00161 Roma, Italy. : Dipartimento MeMoMat, Università di Roma “La Sapienza”, Via A. Scarpa 16, 00161 Roma, Italy. loreti@dmmm.uniroma1.it.
Get access

Abstract

Motivated by a classical work of Erdős we give rather precise necessary and sufficient growth conditions on the nonlinearity in a semilinear wave equation in order to have global existence for all initial data. Then we improve some former exact controllability theorems of Imanuvilov and Zuazua.

Keywords

Wave equationsemilinear equationintegral inequality
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 4 , 1999 , pp. 37 - 56
Copyright
© EDP Sciences, SMAI, 1999

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Cazenave, T. and Haraux, A., Équations d'évolution avec non linéarité logarithmique. Ann. Fac. Sci. Toulouse 2 (1980) 21-51. CrossRef
T. Cazenave and A. Haraux, Introduction aux problèmes d'évolution semi-linéaires. Mathématiques et applications, Vol. 1, Ellipses et SMAI, Paris (1990).
On, P. Erdos the law of the iterated logarithm. Ann. of Math. 43 (1942) 419-436.
Imanuvilov, O.Yu., Boundary control of semilinear evolution equations. Russian Math. Surveys 44 (1989) 183-184.
Li, Ta-Tsien and Bing-Yu Zhang, Global exact controllability of a class of quasilinear hyperbolic systems. J. Math. Anal. Appl. 225 (1998) 289-311. CrossRef
J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod-Gauthier-Villars, Paris (1969).
V.G. Maz'ja, Sobolev Spaces. Springer-Verlag, New York (1985).
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, New York (1983).
S.L. Sobolev, Partial Differential Equations of Mathematical Physics. Dover, New York (1989).
Zuazua, E., Exact controllability for semilinear wave equations in one space dimension. Ann. Inst. H. Poincaré Anal. Non Linéaire 10 (1993) 109-129. CrossRef