Regularity of the solutions of second order evolution equations and their attractors
Regularity results for semilinear and geometric wave equations
Remark on the null-condition for the nonlinear wave equation
Dimostriamo l'esistenza della soluzione globale per un sistema di equazioni delle onde con nonlinearità quadratica dipendente dalle variabili spazio-tempo. Come in [3] la tecnica è basata sulla trasformazione di Penrose.
Remarks on bounded solutions of a semilinear dissipative hyperbolic equation
Remarks on quasilinear evolutions equations.
Remarks on the existence an uniqueness of global decaying solutions of the nonlinear dissipative wave equations.
Remarks on the existence and decay of the nonlinear beam equation.
Remarks on the Klein-Gordon equation
Remarks on the qualitative behavior of the undamped Klein-Gordon equation
We present sufficient conditions on the initial data of an undamped Klein-Gordon equation in bounded domains with homogeneous Dirichlet boundary conditions to guarantee the blow up of weak solutions. Our methodology is extended to a class of evolution equations of second order in time. As an example, we consider a generalized Boussinesq equation. Our result is based on a careful analysis of a differential inequality. We compare our results with the ones in the literature.
Remarks on the wave equation with a nonlinear term with respect to the velocity
Remarks on weak stabilization of semilinear wave equations
If a second order semilinear conservative equation with esssentially oscillatory solutions such as the wave equation is perturbed by a possibly non monotone damping term which is effective in a non negligible sub-region for at least one sign of the velocity, all solutions of the perturbed system converge weakly to 0 as time tends to infinity. We present here a simple and natural method of proof of this kind of property, implying as a consequence some recent very general results of Judith Vancostenoble....
Remarks on weak stabilization of semilinear wave equations
If a second order semilinear conservative equation with esssentially oscillatory solutions such as the wave equation is perturbed by a possibly non monotone damping term which is effective in a non negligible sub-region for at least one sign of the velocity, all solutions of the perturbed system converge weakly to 0 as time tends to infinity. We present here a simple and natural method of proof of this kind of property, implying as a consequence some recent very general results of Judith Vancostenoble. ...
Remarques sur le problème de Cauchy pour l'équation des ondes
Résolution de l’équation où est linéaire et dérive d’un potentiel convexe
Resolvent estimates and the decay of the solution to the wave equation with potential
We prove a weighted estimate for the solution to the linear wave equation with a smooth positive time independent potential. The proof is based on application of generalized Fourier transform for the perturbed Laplace operator and a finite dependence domain argument. We apply this estimate to prove the existence of global small data solution to supercritical semilinear wave equations with potential.
Rigidity for the hyperbolic Monge-Ampère equation
Some properties of nonlinear partial differential equations are naturally associated with the geometry of sets in the space of matrices. In this paper we consider the model case when the compact set is contained in the hyperboloid , where , the set of symmetric matrices. The hyperboloid is generated by two families of rank-one lines and related to the hyperbolic Monge-Ampère equation . For some compact subsets containing a rank-one connection, we show the rigidity property of by imposing...
Rotationally invariant periodic solutions of semilinear wave equations.
Rothe time-discretization method applied to a quasilinear wave equation subject to integral conditions.
Scalar differential invariants of symplectic Monge-Ampère equations
All second order scalar differential invariants of symplectic hyperbolic and elliptic Monge-Ampère equations with respect to symplectomorphisms are explicitly computed. In particular, it is shown that the number of independent second order invariants is equal to 7, in sharp contrast with general Monge-Ampère equations for which this number is equal to 2. We also introduce a series of invariant differential forms and vector fields which allow us to construct numerous scalar differential invariants...
Scattering for Semilinear Wave Equations with Small Data in Three Space Dimensions.