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Qualitative study of three nonlinear parabolic problems

Abstract : This thesis is concerned with the study of three nonlinear parabolic problems : We start with a mathematical model for a micro-electro-mechanical system (MEMS) with variable dielectric permittivity. The model is based on a parabolic equation with singular nonlinearity which describes the dynamic deffection of an elastic plate under the effect of an electrostatic potential. We study the touchdown, or quenching, phenomenon. With the aim of controlling the touchdown set, we give results concerning the touchdownl ocalization in terms of the permittivity profile. In the second part of the thesis, we study a diffusive Hamilton-Jacobi equation in a bounded domain with zero Dirichlet boundary conditions. We analyze the gradient blow-up (GBU) that solutions can exhibit on the boundary of the domain. In a previous work, it was shown that single-point GBU solutions can be constructed in very particular domains, namely, locally fat domains and disks. We prove the existence of this kind ofsolutions for a large family of domains, for which the curvature of the domain may be nonconstant near the GBU point. In the last part of the thesis, we study the evolution problem associated to the j-th eigenvalue of the Hessian matrix. First, we show the existence of a (unique) viscosity solution, which can be approximated by the value function of a two-player zero-sumgame as the step length of the game goes to zero. Then, we show that solutions to this evolution problem converge exponentially fast to the unique stationary solution as t goes to ∞. Finally, we show that in some special cases (for affine boundary data) the solution coincides with the stationary solution in finite time.
Keywords : Hessian matrix
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Submitted on : Monday, January 11, 2021 - 3:53:32 PM
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  • HAL Id : tel-03106136, version 1

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Carlos Esteve Yague. Qualitative study of three nonlinear parabolic problems. Analysis of PDEs [math.AP]. Université Sorbonne Paris Cité, 2019. English. ⟨NNT : 2019USPCD033⟩. ⟨tel-03106136⟩

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