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The Neumann problem for a Barenblatt equation with a multiplicative stochastic force and a nonlinear source term

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Abstract

In this paper, we are interested in an existence and uniqueness result for a Barenblatt’s type equation forced by a multiplicative noise, with additionally a nonlinear source term and under Neumann boundary conditions. The idea to show such a well-posedness result is to investigate in a first step the additive case with a linear source term. Through a time-discretization of the equation and thanks to results on maximal monotone operators, one is able to handle the non-linearity of the equation and pass to the limit on the discretization parameter. This allows us to show existence and uniqueness of a solution in the case of an additive noise and a linear source term. In a second step, thanks to a fixed point procedure, one shows the announced result.

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Notes

  1. For a given separable Hilbert space X, we denote by \({\mathcal {N}}^2_w(0,T,X)\) the space of predictable X-valued processes endowed with the norm \(||\phi ||^2_{{\mathcal {N}}^2_w(0,T,X)}=E\left[ \int _0^T||\phi ||^2_Xdt\right] \) (see Da Prato-Zabczyk [11] p.94).

  2. \({\mathscr {C}}_w\left([0,T], L^{2}(\Omega , H^1(D))\right)\) denotes the set of functions defined on [0, T] with values in \(L^{2}(\Omega , H^1(D))\) which are weakly continuous.

  3. \(W\big (0,T, H^{1}(D), L^{2}(D)\big )=\{v\in L^{2}(0,T, H^{1}(D))\text { such that }\partial _{t}v\in L^{2}(0,T,L^{2}(D))\}\).

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Acknowledgements

The authors wish to thank the joint program between the Higher Education Commission from Pakistan Ministry of Higher Education and the French Ministry of Foreign and European Affairs for the funding of A. Maitlo’s PhD thesis.

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  1. Aix-Marseille Université, CNRS, Centrale Marseille, Laboratoire de Mécanique et d’Acoustique, 4 impasse Nikola Tesla, 13013, Marseille, France

    Caroline Bauzet, Frédéric Lebon & Asghar Maitlo

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  1. Caroline Bauzet
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  2. Frédéric Lebon
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  3. Asghar Maitlo
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Correspondence to Caroline Bauzet.

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Bauzet, C., Lebon, F. & Maitlo, A. The Neumann problem for a Barenblatt equation with a multiplicative stochastic force and a nonlinear source term. Nonlinear Differ. Equ. Appl. 26, 21 (2019). https://doi.org/10.1007/s00030-019-0567-5

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