|Hyperbolic equations with variable nonlinearity: existence and blow-up
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 blow-up of the energy solutions.
Also we consider Young measure solutions of such equations. The analysis relies on the
methods developed in [1-6].
1. Antontsev S.N., Diaz J.I., Shmarev S.I., Energy Methods for Free Boundary Problems : Applications to Non-linear 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: Blow-up of solutions, C.R. Mecanique, 339, 12(2011),751--755.
3. Antontsev S.N., Wave equation with p(x,t)- Laplacian and damping term: Existence and blow-up, Differ. Equ. Appl., 3 (2011), pp.~503--525.
4. Antontsev S.N., Shmarev S., Blow-up of solutions to parabolic equations with nonstandard growth conditions. J. Comput. Appl. Math., 234 (2010), pp.2633--2645.
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.153-167.
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.62-77.