Explosion en temps fini de solutions d’équations dispersives ou dissipatives non-linéaires

Abstract : The subject of this thesis is the formation of singularities for some nonlinear evolution equations of dissipative and/or dispersive type. Our work is focused on the Cauchy problems, usually with periodic boundary conditions or on the whole $\mathbb{R}^{n}$. Our aim is to provide the necessary or sufficient conditions (or both) on the initial data $u_0 (x)$, ensuring that the lifetime $T^{*}$ of the solution resulting from $u_0$ is finite or not. We study two types of equations: a nonlinear parabolic equation and a class of dispersive wave equations. In the first case, we study a one-dimensional model which describe the propagation of nonlinear waves in a channel or the deformations of a hyper-elastic rod. One decisive contibutions of our work will be this: the only global strong periodic solution of the rod equation vanishing in at least one point is the identically zero solution. We also establish the analogue of this result in the case of non-periodic solutions defined on the whole real line which vanish at infinity. Our analysis is based on the application of new local-in-space blowup criteria. The second equation that we consider is a generalization of the rod equation which was proposed by H. Holden and X. Raynaud. This generalization covers many other equations with interesting mathematical properties. We will establish criteria for the blowup in finite time that involve only the properties of the data $u_0$ in a neighborhood of a single point, thus simplifying and extending earlier blowup criteria for this equation. After, we study family of equations known in the literature as the $b$-family equations. One of the most notable cases of this family of equations is the Degasperis-Procesi equation. For this family we obtain similar results as those described above. Finally, the last part, we study the well-posedness, locally or globally in time of the nonlinear heat equation, in functional spaces having appropriate invariance properties relative to scale changes. After extending Y. Meyer's result establishing the existence of global solutions, under a smallness condition of the initial data in the homogeneous Besov spaces $\dot{B}_{p}^{-\sigma, \infty}(\mathbb{R}^{3})$, where $3 < p < 9$ and $\sigma=1-3/p$, we prove that initial data $u_0\in \mathcal{S}(\mathbb{R}^{3})$, arbitrarily small in ${\dot B^{-2/3,\infty}_{9}}(\mathbb{R}^{3})$, can produce solutions that explode in finite time. In addition, the blowup may occur after an arbitrarily short time.
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Equations aux dérivées partielles [math.AP]. Université Claude Bernard - Lyon I, 2015. Français. 〈NNT : 2015LYO10198〉
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Manuel Fernando Cortez. Explosion en temps fini de solutions d’équations dispersives ou dissipatives non-linéaires. Equations aux dérivées partielles [math.AP]. Université Claude Bernard - Lyon I, 2015. Français. 〈NNT : 2015LYO10198〉. 〈tel-01814003〉

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