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Well-posedness for nonlinear SPDEs with strongly continuous perturbation

Part of: Parabolic equations and systems Stochastic analysis

Published online by Cambridge University Press:  11 March 2020

Guy Vallet  and
Aleksandra Zimmermann
Guy Vallet
Affiliation:
LMAP UMR CNRS 5142, IPRA BP 1155, 64013Pau Cedex, France ([email protected])
Aleksandra Zimmermann
Affiliation:
Faculty of Mathematics, Thea-Leymann-Str. 9, 45127Essen, Germany ([email protected])
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Abstract

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We consider the well-posedness of a stochastic evolution problem in a bounded Lipschitz domain D ⊂ ℝd with homogeneous Dirichlet boundary conditions and an initial condition in L2(D). The main technical difficulties in proving the result of existence and uniqueness of a solution arise from the nonlinear diffusion-convection operator in divergence form which is given by the sum of a Carathéodory function satisfying p-type growth associated with coercivity assumptions and a Lipschitz continuous perturbation. In particular, we consider the case 1 < p < 2 with an appropriate lower bound on p determined by the space dimension. Another difficulty arises from the fact that the additive stochastic perturbation with values in L2(D) on the right-hand side of the equation does not inherit the Sobolev spatial regularity from the solution as in the multiplicative noise case.

Keywords

Nonlinear SPDEstrongly continuous perturbationadditive stochastic forcingpseudomonotone SPDE

MSC classification

Primary: 60H15: Stochastic partial differential equations
Secondary: 35K92: Quasilinear parabolic equations with $p$-Laplacian 35K55: Nonlinear parabolic equations
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 151 , Issue 1 , February 2021 , pp. 265 - 295
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press

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