Dynamique des équations des ondes avec amortissement variable

Abstract : This thesis concerns the qualitative study of the dynamics of the damped wave equations on a bounded domain Ω of R^d. We begin with a chapter describing the various notions of stability and reviewing the known results.
In the next chapter, we prove the genericity, with respect to the non-linearity, of the Morse-Smale property for the one-dimensional wave equation with internal damping γ(x) (WEID) and the one with boundary damping g(x)δ_{x on the boundary} (WEBD). The proof mainly uses accurate properties of the asymptotic behaviour of the functions t--->u(x_0,t), where u is a bounded solution of either Equations (WEID) or (WEBD), or their adjoint equations, and where x_0 is a fixed point of Ω. This asymptotic behaviour is deduced from the spectral properties of the linearized operator around an equilibrium point. In particular, the eigenvectors of this operator form a Riesz basis and its eigenvalues are generically simple.
The last part of this thesis concerns the study of the convergence of the dynamics of Equation (WEID) to the ones of Equation (WEBD) when the sequence of internal dampings γ_n(x) converges to g(x)δ_{x on the boundary} in the sense of distributions. In dimension d=1, we show that the dynamics of (WEID) converge to the ones of (WEBD). In dimension d>1, weaker results of convergence of attractors are obtained. The perturbation studied here is singular and thus, some classical results of stability have to be generalized. To obtain the best results of convergence, one has to show that the linear semigroups associated with (WEID) decay with an exponential rate ||e^{A_nt}||X
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Romain Joly. Dynamique des équations des ondes avec amortissement variable. Mathématiques [math]. Université Paris Sud - Paris XI, 2005. Français. ⟨tel-00011715⟩

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