The Neumann problem for a Barenblatt equation with a multiplicative stochastic force and a nonlinear source term
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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.
KeywordsStochastic Barenblatt equation Multiplicative noise Additive noise Stochastic force Itô integral Maximal monotone operator Neumann condition Time discretization Heat equation Fixed point
Mathematics Subject ClassificationPrimary 47J35 60H15 Secondary 47H10 47H05
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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