# Well posedness and control of semilinear wave equations with iterated logarithms

Piermarco Cannarsa; Vilmos Komornik; Paola Loreti

ESAIM: Control, Optimisation and Calculus of Variations (2010)

- Volume: 4, page 37-56
- ISSN: 1292-8119

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topCannarsa, Piermarco, Komornik, Vilmos, and Loreti, Paola. "Well posedness and control of semilinear wave equations with iterated logarithms." ESAIM: Control, Optimisation and Calculus of Variations 4 (2010): 37-56. <http://eudml.org/doc/197290>.

@article{Cannarsa2010,

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.
},

author = {Cannarsa, Piermarco, Komornik, Vilmos, Loreti, Paola},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Wave equation; semilinear equation; integral inequality; global existence for all initial data; exact controllability},

language = {eng},

month = {3},

pages = {37-56},

publisher = {EDP Sciences},

title = {Well posedness and control of semilinear wave equations with iterated logarithms},

url = {http://eudml.org/doc/197290},

volume = {4},

year = {2010},

}

TY - JOUR

AU - Cannarsa, Piermarco

AU - Komornik, Vilmos

AU - Loreti, Paola

TI - Well posedness and control of semilinear wave equations with iterated logarithms

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2010/3//

PB - EDP Sciences

VL - 4

SP - 37

EP - 56

AB -
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.

LA - eng

KW - Wave equation; semilinear equation; integral inequality; global existence for all initial data; exact controllability

UR - http://eudml.org/doc/197290

ER -

## References

top- T. Cazenave and A. Haraux, Équations d'évolution avec non linéarité logarithmique. Ann. Fac. Sci. Toulouse2 (1980) 21-51.
- T. Cazenave and A. Haraux, Introduction aux problèmes d'évolution semi-linéaires. Mathématiques et applications, Vol. 1, Ellipses et SMAI, Paris (1990).
- P. Erdos, On the law of the iterated logarithm. Ann. of Math.43 (1942) 419-436.
- O.Yu. Imanuvilov, Boundary control of semilinear evolution equations. Russian Math. Surveys44 (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.
- 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).
- E. Zuazua, Exact controllability for semilinear wave equations in one space dimension. Ann. Inst. H. Poincaré Anal. Non Linéaire10 (1993) 109-129.

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