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1 results

  • Well-posedness for nonlinear SPDEs with strongly continuous perturbation

  • Part of
  • Guy Vallet, Aleksandra Zimmermann
  • Journal:
    Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 151 / Issue 1 / February 2021
    Published online by Cambridge University Press:
    11 March 2020, pp. 265-295
    Print publication:
    February 2021
    • Article
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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.