Using HUM, we study the problem of exact controllability with Neumann boundary conditions for second order hyperbolic equations. We prove that these systems are exactly controllable for all initial states in $L^2\left(\Omega \right)\times {\left(H^1\left(\Omega \right)\right)}^{\text{'}}$ and we derive estimates for the control time T.

Le but de l’exposé est de donner un guide de lecture pour un article de Gilles Lebeau où il est montré que le problème de Cauchy pour l’équation d’onde surcritique $({\partial }_t^2-{\Delta }_x)u+u^p=0$ est mal posé au sens de Hadamard dans l’espace d’énergie, pour $p\ge 7$ en dimension 3. La preuve repose sur des constructions d’optique géométrique et des analyses d’instabilité dans des régimes fortement non linéaires. On donnera les étapes de l’analyse en essayant de les situer dans leur contexte plus général : construction de solutions...

The present paper studies the existence and uniqueness of global solutions and decay rates to a given nonlinear hyperbolic problem.

We prove existence and asymptotic behaviour of a weak solutions of a mixed problem for $$\begin{array}{cc}& u^{\text{'}\text{'}}+Au-\Delta u^{\text{'}}+{\left|v\right|}^{\rho +2}{\left|u\right|}^{\rho }u=f_1\hfill \\ & v^{\text{'}\text{'}}+Av-\Delta v^{\text{'}}+{\left|u\right|}^{\rho +2}{\left|v\right|}^{\rho }v=f_2*\hfill \end{array}$$ where $A$ is the pseudo-Laplacian operator.

We consider the damped semilinear viscoelastic wave equation $$u^{\text{'}\text{'}}-\Delta u+{\int }_0^{t}h(t-\tau )div\left\{a\nabla u\left(\tau \right)\right\}\mathrm{d}\tau +g\left(u^{\text{'}}\right)=0\text{in}\Omega \times (0,\infty )$$ with nonlocal boundary dissipation. The existence of global solutions is proved by means of the Faedo-Galerkin method and the uniform decay rate of the energy is obtained by following the perturbed energy method provided that the kernel of the memory decays exponentially.

The existence and uniqueness of classical global solution and blow up of non-global solution to the first boundary value problem and the second boundary value problem for the equation $$u_{tt}-\alpha u_{xx}-\beta u_{xxtt}=\varphi {\left(u_x\right)}_x$$ are proved. Finally, the results of the above problem are applied to the equation arising from nonlinear waves in elastic rods $$u_{tt}-\left[a_0+n{a}_1{\left(u_x\right)}^{n-1}\right]u_{xx}-a_2{u}_{xxtt}=0.$$

Problems of existence and nonexistence of global nontrivial solutions to quasilinear evolution differential inequalities in a product of cones are investigated. The proofs of the nonexistence results are based on the test-function method developed, for the case of the whole space, by Mitidieri, Pohozaev, Tesei and Véron. The existence result is established using the method of supersolutions.

We consider the initial-boundary value problem for a nonlinear higher-order nonlinear hyperbolic equation in a bounded domain. The existence of global weak solutions for this problem is established by using the potential well theory combined with Faedo-Galarkin method. We also established the asymptotic behavior of global solutions as $t\to \infty$ by applying the Lyapunov method.