A bifurcation theory for periodic solutions of nonlinear dissipative hyperbolic equations
A boundary value problem for quasilinear hyperbolic systems with a retarded argument
A Carleman estimates based approach for the stabilization of some locally damped semilinear hyperbolic equations
First, we consider a semilinear hyperbolic equation with a locally distributed damping in a bounded domain. The damping is located on a neighborhood of a suitable portion of the boundary. Using a Carleman estimate [Duyckaerts, Zhang and Zuazua, Ann. Inst. H. Poincaré Anal. Non Linéaire (to appear); Fu, Yong and Zhang, SIAM J. Contr. Opt. 46 (2007) 1578–1614], we prove that the energy of this system decays exponentially to zero as the time variable goes to infinity. Second, relying on another Carleman...
A Carleman estimates based approach for the stabilization of some locally damped semilinear hyperbolic equations
First, we consider a semilinear hyperbolic equation with a locally distributed damping in a bounded domain. The damping is located on a neighborhood of a suitable portion of the boundary. Using a Carleman estimate [Duyckaerts, Zhang and Zuazua, Ann. Inst. H. Poincaré Anal. Non Linéaire (to appear); Fu, Yong and Zhang, SIAM J. Contr. Opt.46 (2007) 1578–1614], we prove that the energy of this system decays exponentially to zero as the time variable goes to infinity. Second, relying on another Carleman...
A Commuting Vectorfields Approach to Strichartz type Inequalities and Applications to Quasilinear Wave Equations
A Dynamic Frictionless Contact Problem with Adhesion and Damage
We consider a dynamic frictionless contact problem for a viscoelastic material with damage. The contact is modeled with normal compliance condition. The adhesion of the contact surfaces is considered and is modeled with a surface variable, the bonding field, whose evolution is described by a first order differential equation. We establish a variational formulation for the problem and prove the existence and uniqueness of the solution. The proofs are based on the theory of evolution equations with...
A hyperbolic problem with nonlinear second-order boundary damping.
A KAM-theorem for some nonlinear partial differential equations
A monotonicity method for solving hyperbolic problems with hysteresis
A mountain pass method for the numerical solution of semilinear wave equations.
A necessary and sufficient condition for the existence of an exponential attractor
We give a necessary and sufficient condition for the existence of an exponential attractor. The condition is formulated in the context of metric spaces. It also captures the quantitative properties of the attractor, i.e., the dimension and the rate of attraction. As an application, we show that the evolution operator for the wave equation with nonlinear damping has an exponential attractor.
A new approach to solve a nonlinear wave equation.
A new method to obtain decay rate estimates for dissipative systems with localized damping.
We consider the wave equation damped with a locally distributed nonlinear dissipation. We improve several earlier results of E. Zuazua and of M. Nakao in two directions: first, using the piecewise multiplier method introduced by K. Liu, we weaken the usual geometrical conditions on the localization of the damping. Then thanks to some new nonlinear integral inequalities, we eliminate the usual assumption on the polynomial growth of the feedback in zero and we show that the energy of the system decays...
A new numerical model for propagation of tsunami waves
A new model for propagation of long waves including the coastal area is introduced. This model considers only the motion of the surface of the sea under the condition of preservation of mass and the sea floor is inserted into the model as an obstacle to the motion. Thus we obtain a constrained hyperbolic free-boundary problem which is then solved numerically by a minimizing method called the discrete Morse semi-flow. The results of the computation in 1D show the adequacy of the proposed model.
A nonexistence result for a nonlinear PDE with Robin condition.
A nonlinear transmission problem with time dependent coefficients.
A nonlinear wave equation with a nonlinear integral equation involving the boundary value.
A nonlocal mixed semilinear problem for second-order hyperbolic equations.
A note on a theorem of Jörgens.
A Note on Maximal Rate of Decay for Solutions to Some Linear Wave Equations.