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Études de problèmes aux limites non linéaires de type pseudo-parabolique

Abstract : The aim of this work is to study the general nonlinear pseudo parabolic problem: find a real valued measurable function $u$ on $Q:=]0,T[\times \Omega$ solution to \begin{equation*} \left\{ \begin{array}{l@{\quad}l} f\left(t,x,u_t\right)-Div \left\{a\left(x,u,u_t\right)\nabla u+b\left(x,u,u_t\right)\nabla u_t \right\}=g(t,x), \; (t,x)\in Q, \\ u(x,t)=0,\; (t,x)\in ]0,T[\times \partial \Omega, \\ u(0,x)=u_0, \; x\in \Omega,\\ \end{array} \right. \end{equation*} where the operator of Nemestki associated to the function $f$ is monotone.\\ In a first chapter, one shows the existence of a solution to the above problem by the way of the implicit time-semidiscretization. The existence of the solutions of the discretized equation is based on the Schauder-Tikhonov fixed point theorem and the convergence of the scheme on an adapted compactness argument. At the end of the chapter, one derives some applications to the equation of Barenblatt and to multivoque $f$. In a second chapter, one is interested in the pseudoparabolic problem of Barenblatt: find a real valued measurable function $u$ such that \begin{equation*} \left\{ \begin{array}{l@{\quad}l} f\left(u_t\right(t,x))-\Delta u(t,x)-\epsilon \Delta u_t(t,x)=g(t,x), \; (t,x)\in Q, \\ u(x,t)=0,\; (t,x)\in ]0,T[\times \partial \Omega, \\ u(0,x)=u_0, \; x\in \Omega,\\ \end{array} \right. \end{equation*} where $f$ is not necessarily monotone.\\ For $\epsilon> \epsilon_0>0 $, where $\epsilon_0$ is a critical value, this problem is well posed by using similar arguments. For the critical value, the problem admits at most a solution. Then, the solution exists with some more assumptions on $f$. When $0<\epsilon<\epsilon_0$, the solution is not unique in general. One proposes finally a stochastic approach to the pseudoparabolic equation of Barenblatt-Sobolev. The last chapter concerns monodimensional simulations, in particular one is interested in the pseudoparabolic singular perturbation when the molecular diffusion changes sign.
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Submitted on : Tuesday, October 5, 2010 - 8:20:15 PM
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Ngonn Seam. Études de problèmes aux limites non linéaires de type pseudo-parabolique. Mathématiques [math]. Université de Pau et des Pays de l'Adour, 2010. Français. ⟨NNT : 10PAUU305⟩. ⟨tel-00523633⟩

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