Well-posedness for nonlinear SPDEs with strongly continuous perturbation

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 nonli...

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Bibliographic Details
Published in:Proceedings of the Royal Society of Edinburgh. Section A. Mathematics 2021-02, Vol.151 (1), p.265-295
Main Authors: Vallet, Guy, Zimmermann, Aleksandra
Format: Article
Language:English
Subjects:
Boundary conditions
Coercivity
Dirichlet problem
Divergence
Lower bounds
Mathematics
Operators (mathematics)
Perturbation
Well posed problems
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Summary: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.
ISSN:0308-2105
1473-7124
DOI:10.1017/prm.2020.13