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Stochastic quasilinear parabolic equations with non standard growth : weak and strong solutions
Thesis (PhD)--University of Pretoria, 2015.
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| Format: | Thesis |
| Language: | English |
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University of Pretoria
2016
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| _version_ | 1867613629104259072 |
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| access_status_str | Open Access |
| author2 | Sango, Mamadou |
| author_browse | Sango, Mamadou |
| author_facet | Sango, Mamadou |
| collection | Thesis |
| dc_rights_str_mv | © 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. |
| description | Thesis (PhD)--University of Pretoria, 2015. |
| format | Thesis |
| id | oai:repository.up.ac.za:2263/53502 |
| institution | University of Pretoria (South Africa) |
| language | English |
| last_indexed | 2026-06-10T12:39:11.002Z |
| license_str | Other — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from UPSpace — University of Pretoria Institutional Repository |
| publishDate | 2016 |
| publishDateRange | 2016 |
| publishDateSort | 2016 |
| publisher | University of Pretoria |
| publisherStr | University of Pretoria |
| record_format | dspace |
| source_str | UPSpace — University of Pretoria Institutional Repository |
| spelling | oai:repository.up.ac.za:2263/53502 Stochastic quasilinear parabolic equations with non standard growth : weak and strong solutions Sango, Mamadou zakaria@aims.ac.za Ali, Zakaria Idriss UCTD Thesis (PhD)--University of Pretoria, 2015. 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. Mathematics and Applied Mathematics PhD Unrestricted 2016-07-01T10:33:07Z 2016-07-01T10:33:07Z 2016-04-13 2015 Thesis 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> A2016 http://hdl.handle.net/2263/53502 en © 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. application/pdf University of Pretoria |
| spellingShingle | UCTD Stochastic quasilinear parabolic equations with non standard growth : weak and strong solutions |
| title | Stochastic quasilinear parabolic equations with non standard growth : weak and strong solutions |
| title_full | Stochastic quasilinear parabolic equations with non standard growth : weak and strong solutions |
| title_fullStr | Stochastic quasilinear parabolic equations with non standard growth : weak and strong solutions |
| title_full_unstemmed | Stochastic quasilinear parabolic equations with non standard growth : weak and strong solutions |
| title_short | Stochastic quasilinear parabolic equations with non standard growth : weak and strong solutions |
| title_sort | stochastic quasilinear parabolic equations with non standard growth weak and strong solutions |
| topic | UCTD |
| url | http://hdl.handle.net/2263/53502 |