Stochastic quasilinear parabolic equations with non standard growth : weak and strong solutions

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dc.contributor.advisor Sango, Mamadou en
dc.contributor.postgraduate Ali, Zakaria Idriss en
dc.date.accessioned 2016-07-01T10:33:07Z
dc.date.available 2016-07-01T10:33:07Z
dc.date.created 2016-04-13 en
dc.date.issued 2015 en
dc.description Thesis (PhD)--University of Pretoria, 2015. en
dc.description.abstract This thesis consists of two main parts. The rst part concerns the existence of weak probabilistic solutions (called elsewhere martingale solutions) for a stochastic quasilinear parabolic equation of generalized polytropic ltration, characterized by the presence of a nonlinear elliptic part admitting nonstandard growth. The deterministic version of the equation was rst introduced and studied by Samokhin in [178] as a generalized model for polytropic ltration. Our objective is to investigate the corresponding stochastic counterpart in the functional setting of generalized Lebesgue and Sobolev spaces. We establish an existence result of weak probabilistic solutions when the forcing terms do not satisfy Lipschitz conditions and the noise involves cylindrical Wiener processes. The second part is devoted to the existence and uniqueness results for a class of strongly nonlinear stochastic parabolic partial di erential equations. This part aims to treat an important class of higher-order stochastic quasilinear parabolic equations involving unbounded perturbation of zeroth order. The deterministic case was studied by Brezis and Browder (Proc. Natl. Acad. Sci. USA, 76(1): 38-40, 1979). Our main goal is to provide a detailed study of the corresponding stochastic problem. We establish the existence of a probabilistic weak solution and a unique strong probabilistic solution. The main tools used in this part of the thesis are a regularization through a truncation procedure which enables us to adapt the work of Krylov and Rozosvkii (Journal of Soviet Mathematics, 14: 1233-1277, 1981), combined with analytic and probabilistic compactness results (Prokhorov and Skorokhod Theorems), the theory of pseudomonotone operators, and a Banach space version of Yamada-Watanabe's theorem due to R ockner, Schmuland and Zhang. The study undertaken in this thesis is in some sense pioneering since both classes of stochastic partial di erential equations have not been the object of previous investigation, to the best of our knowledge. The results obtained are therefore original and constitute in our view signi cant contribution to the nonlinear theory of stochastic parabolic equations. en
dc.description.availability Unrestricted en
dc.description.degree PhD en
dc.description.department Mathematics and Applied Mathematics en
dc.identifier.citation Ali, ZI 2016, Stochastic quasilinear parabolic equations with non standard growth : weak and strong solutions, PhD Thesis, University of Pretoria, Pretoria, viewed yymmdd <http://hdl.handle.net/2263/53502> en
dc.identifier.other A2016 en
dc.identifier.uri http://hdl.handle.net/2263/53502
dc.language.iso en en
dc.publisher University of Pretoria en_ZA
dc.rights © 2016, University of Pretoria. All rights reserved. The copyright in this work vests in the University of Pretoria. No part of this work may be reproduced or transmitted in any form or by any means, without the prior written permission of the University of Pretoria. en
dc.subject UCTD en
dc.title Stochastic quasilinear parabolic equations with non standard growth : weak and strong solutions en
dc.type Thesis en


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