An order approach to SPDEs with antimonotone terms
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An order approach to SPDEs with antimonotone terms Luca Scarpa1
· Ulisse Stefanelli1,2
Received: 29 October 2019 / Revised: 6 December 2019 © The Author(s) 2020
Abstract We consider a class of parabolic stochastic partial differential equations featuring an antimonotone nonlinearity. The existence of unique maximal and minimal variational solutions is proved via a fixed-point argument for nondecreasing mappings in ordered spaces. This relies on the validity of a comparison principle. Keywords Existence · Parabolic SPDEs · Antimonotone term · Comparison principle · Order methods Mathematics Subject Classification 35K55 · 35R60 · 60H15
1 Introduction This note is concerned with the existence of solutions to a class of parabolic stochastic partial differential equations (SPDEs). The typical setting that we have in mind is the equation du − div(a(∇u)) dt − b(u) dt = f (u) dt + G(u) dW in (0, T ) × O
(1.1)
suitably coupled with boundary and initial conditions, with O being a smooth bounded domain of Rd and T > 0 a fixed final time. Here, the real-valued variable u is defined on × [0, T ] × O, a is monotone and polynomial, f is Lipschitz continuous, and G is a Lipschitz-type operator, stochas-
B
Luca Scarpa [email protected] http://www.mat.univie.ac.at/∼scarpa Ulisse Stefanelli [email protected] http://www.mat.univie.ac.at/∼stefanelli
1
Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria
2
Istituto di Matematica Applicata e Tecnologie Informatiche E. Magenes, v. Ferrata 1, 27100 Pavia, Italy
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Stoch PDE: Anal Comp
tically integrable with respect to W , a cylindrical Wiener process on the underlying probability space (, F , P). The function b : R → R is nondecreasing, possibly being nonsmooth, so that the corresponding term in the left-hand side of the equation is indeed antimonotone. Our aim is to prove that a variational formulation of relation (1.1) admits a solution, whenever complemented with suitable initial and boundary conditions. If b is Lipschitz continuous or −b is nondecreasing and continuous such existence follows from the classical theory by Pardoux [13] and Krylov-Rozovski˘ı [8], see also Liu-Röckner [11]. By contrast, we focus here in the case of b linearly bounded but not continuous nor nondecreasing. This situation, to the best of our knowledge, has yet to be addressed. Indeed, the possible discontinuity of − b prevents it from being even locally Lipschitz-continuous, hence also the refined well-posedness results for SPDEs with locally monotone or locally Lipschitz-continuous drift (see again [11]) cannot be applied. The case of a nondecreasing but not Lipschitz continuous nonlinearity b in (1.1) prevents from proving existence by a standard regularization or approximation approach. In fact, the usual parabolic compactness seem to be of little use in order to pass to the limit in the antimonotone term − b(u). We resort here in tackling the problem in an ordered-space framework instead, by exploiting the fact that b is nondec
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