Equations d'évolution sur certains groupes hyperboliques

Abstract : This thesis focuses on the study of evolution equations on certain hyperbolic groups, in particular, we study the heat equation, the Schrödinger equation and the modified wave equation first on homogeneous trees then on symmetric graphs. In the homogeneous trees case, we show that under a gauge invariance condition, we have global existence of solutions of the Schrödinger equation and scattering for arbitrary data in the space of square integrable functions without any restriction on the degree of the nonlinearity, in contrast to the euclidean and hyperbolic space cases. We then generalize this result on symmetric graphs of degree (k − 1)(r − 1) under the condition k < r . One of our main results on symmetric graphs is the estimate of the heat kernel associated to the combinatorial laplacian. Finally, we establish an explicit expression of solutions of the modified wave equation on symmetric graphs.
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Alaa Jamal Eddine. Equations d'évolution sur certains groupes hyperboliques. Mathématiques générales [math.GM]. Université d'Orléans, 2013. Français. ⟨NNT : 2013ORLE2055⟩. ⟨tel-01022926⟩

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