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Finite time singularity formation for non symmetric or non variational partial differential equations

Abstract : In the context of this thesis, we are interested in finite time singularity formation for non symmetric or non variational partial differential equations of parabolic type. In particular, we mainly focus on the following two phenomena : blowup and quenching (touch-down) infinite time. In this thesis, we aim at studying the following equations : [....] where Ω is a C² bounded domain in ℝᶰ and λ, Ƴ are positive constants.These models are closely related to many common phenomena in nature. In particular, equation (6) is a model for Micro Electro Mechanical Systems (MEMS). In this work, we construct blowup solutions to (4) and (5) and solutions with extinction to (6). In addition to that, we describe the asymptotic behavior of these solutions around the singular point. We use in this thesis the framework of similarity variables, introduced by Giga and Kohn in CPAM 1985. We finally derive the results by using a reduction to a finite dimensional problem and a topological argument which was introduced in particular by Bressan, Bricmont and Kupiainen, and also Merle and Zaag. Clearly, our work is not a simple adaptation of the works cited above. Indeed, our models, by their proximity to applications, are outside the ideal framework considered in pioneering works. In particular, equation (4) is not scaling-invariant, whereas (5) does not admit variational structure. As for (6), the presence of the integral term (non-local term) requires us to treat this term more delicately. In fact, we have achieved our goals thanks to some new ideas. More precisely, for (5), we carry out a delicate control of the solution so that it always stays in the domain where the non linearity is defined with no ambiguity. For (6), we control the oscillation of the non-local term to keep it small enough, and this allows us to deduce its convergence.
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Giao Ky Duong. Finite time singularity formation for non symmetric or non variational partial differential equations. Analysis of PDEs [math.AP]. Université Sorbonne Paris Cité, 2019. English. ⟨NNT : 2019USPCD058⟩. ⟨tel-03186715⟩

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