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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 $(\partial_{t}^{2} - Δ_{x}) u + u^{p} = 0$ est mal posé au sens de Hadamard dans l’espace d’énergie, pour $p \geq 7$ 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...

Existence and asymptotic behaviour of solutions of a nonlinear evolution problem

Nelson Nery Oliveira Castro (1997)

Applications of Mathematics

We prove existence and asymptotic behaviour of a weak solutions of a mixed problem for $\begin{matrix} u^{''} + A u - Δ u^{'} + {| v |}^{ρ + 2} {| u |}^{ρ} u = f_{1} \\ v^{''} + A v - Δ v^{'} + {| u |}^{ρ + 2} {| v |}^{ρ} v = f_{2} * \end{matrix}$ where $A$ is the pseudo-Laplacian operator.

Existence and asymptotic expansion of solutions to a nonlinear wave equation with a memory condition at the boundary.

Nguyen Thanh Long, Le Xuan Truong (2007)

Electronic Journal of Differential Equations (EJDE) [electronic only]

Existence and asymptotic stability for viscoelastic problems with nonlocal boundary dissipation

Jong Yeoul Park, Sun Hye Park (2006)

Czechoslovak Mathematical Journal

We consider the damped semilinear viscoelastic wave equation $u^{''} - Δ u + \int_{0}^{t} h (t - τ) div {a \nabla u (τ)} d τ + g (u^{'}) = 0 in Ω \times (0, \infty)$ 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.

Existence and nonexistence results for reaction-diffusion equations in product of cones

Abdallah Hamidi, Gennady Laptev (2003)

Open Mathematics

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.

Existence and uniqueness of solutions of nonlocal problems for hyperbolic equation $u_{x t} = F (x, t, u, u_{x})$ .

Byszewski, L. (1990)

Journal of Applied Mathematics and Stochastic Analysis

Existence and uniqueness of solutions to wave equations with nonlinear degenerate damping and source terms

Viorel Barbu, Irena Lasiecka, Mohammad Rammaha (2005)

Control and Cybernetics

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