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Hyperbolic equations with variable nonlinearity: existence and blowup
S. Antontsev
CMAF, University of Lisbon, Portugal
We study the Dirichlet problem for a class of nonlinear hyperbolic equations
with p(x,t) Laplacian relatively spatial variables and with damping term.
Under suitable conditions on the data, we prove local and global existence
theorems and study the finite time blowup of the energy solutions.
Also we consider Young measure solutions of such equations. The analysis relies on the
methods developed in [16].
References
1. Antontsev S.N., Diaz J.I., Shmarev S.I., Energy Methods for Free Boundary Problems : Applications to Nonlinear PDEs and Fluid Mechanics. Bikhauser, Boston, 2002. Progress in Nonlinear Differential Equations and Their Applications, Vol. 48.
2. Antontsev S.N., Wave equation with p(x,t)Laplacian and damping term: Blowup of solutions, C.R. Mecanique, 339, 12(2011),751755.
3. Antontsev S.N., Wave equation with p(x,t) Laplacian and damping term: Existence and blowup, Differ. Equ. Appl., 3 (2011), pp.~503525.
4. Antontsev S.N., Shmarev S., Blowup of solutions to parabolic equations with nonstandard growth conditions. J. Comput. Appl. Math., 234 (2010), pp.26332645.
5. Antontsev S., Amorim P., Young measure solutions for wave equation with p(x,t)Laplacian: Existence and blow up, Nonlinear Analysis: Theory, Methods and Applications, 92(2013), pp.153167.
6. Antontsev S., Ferreira J., Existence, uniqueness and blow up for hyperbolic equations with nonstandard growth conditions, Nonlinear Analysis Series A: Theory, Methods and Applications, 93(2013), pp.6277. 
