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

Abstract

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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.

How to cite

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Cannarsa, 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

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  1. T. Cazenave and A. Haraux, Équations d'évolution avec non linéarité logarithmique. Ann. Fac. Sci. Toulouse2 (1980) 21-51.  Zbl0411.35051
  2. T. Cazenave and A. Haraux, Introduction aux problèmes d'évolution semi-linéaires. Mathématiques et applications, Vol. 1, Ellipses et SMAI, Paris (1990).  Zbl0786.35070
  3. P. Erdos, On the law of the iterated logarithm. Ann. of Math.43 (1942) 419-436.  Zbl0063.01264
  4. O.Yu. Imanuvilov, Boundary control of semilinear evolution equations. Russian Math. Surveys44 (1989) 183-184.  
  5. 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.  Zbl0915.93007
  6. J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod-Gauthier-Villars, Paris (1969).  
  7. V.G. Maz'ja, Sobolev Spaces. Springer-Verlag, New York (1985).  
  8. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, New York (1983).  Zbl0516.47023
  9. S.L. Sobolev, Partial Differential Equations of Mathematical Physics. Dover, New York (1989).  Zbl0123.06508
  10. E. Zuazua, Exact controllability for semilinear wave equations in one space dimension. Ann. Inst. H. Poincaré Anal. Non Linéaire10 (1993) 109-129.  Zbl0769.93017

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