Abstract
In this paper, we discuss the existence and uniqueness of variational solutions for a stochastic heat equation in a bounded domain with both additive and laminar multiplicative noise and with a nonlinear dissipative condition on the boundary. Our construction includes, as a particular case, the Signorini-type boundary conditions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
Recall that from (3) we have
$$\begin{aligned} \beta _\lambda (u) u \ge \alpha _3 |u|^{m+1} + \alpha _4, \end{aligned}$$so we get
$$\begin{aligned} \beta _\lambda (u) u&= \beta (J_\lambda (u)) J_\lambda (u) - \beta (J_\lambda (u)) [u - J_\lambda (u)] = \beta (J_\lambda (u)) J_\lambda (u) - \lambda |\beta (J_\lambda (u))|^2\\&\ge \alpha _3 |J_\lambda (u)|^{m+1} + \alpha _4 - \lambda |\beta (J_\lambda (u))|^2 \ge \alpha _3 |J_\lambda (u)|^{m+1} + \alpha _4 \end{aligned}$$
References
Barbu, V., Da Prato, G., Röckner, M.: Stochastic porous media equations and self-organized criticality. Comm. Math. Phys. 285(3), 901–923 (2009)
Barbu, V., Röckner, M.: On a random scaled porous media equation. J. Differen. Equ. 251(9), 2494–2514 (2011)
Barbu, V., Röckner, M.: Stochastic variational inequalities and applications to the total variation flow perturbed by linear multiplicative noise. Arch. Ration. Mech. Anal. 209(3), 797–834 (2013)
Krylov, N.V., Rozovskiĭ, B.L.: Stochastic evolution equations. In Current Problems in Mathematics, vol. 14, pp. 71–147, 256 (1979). (Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Informatsii, Moscow, Russian)
Prévôt, C., Röckner, M.: A Concise Course on Stochastic Partial Differential Equations. Lecture Notes in Mathematics, vol. 1905. Springer, Berlin (2007)
Duvaut, G., Lions, J.-L.: Inequalities in Mechanics and Physics. Springer, Berlin (1976). Translated from the French by C. W. John Grundlehren der Mathematischen Wissenschaften
Fichera, G.: Problemi elastostatici con vincoli unilaterali: il problema di Signorini con ambigue condizioni al contorno. Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. Sez. I 8(7), 91–140 (1963/1964)
Brézis, H.: Problèmes unilatéraux. J. Math. Pures Appl. 9(51), 1–168 (1972)
Barbu, V.: Nonlinear Differential Equations of Monotone Types in Banach Spaces. Springer Monographs in Mathematics. Springer, New York (2010)
Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions, 2nd edn. Cambridge University Press, Cambridge (2014)
Barbu, V., Röckner, M., Zhang, D.: Stochastic nonlinear Schrödinger equations with linear multiplicative noise: rescaling approach. J. Nonlinear Sci. 24, 383–409 (2014)
Barbu, V., Bonaccorsi, S., Tubaro, L.: A stochastic parabolic equation with nonlinear flux on the boundary driven by a Gaussian noise. SIAM J. Math. Anal. 46(1), 780–802 (2014)
Barbu, V., Bonaccorsi, S., Tubaro, L.: Stochastic parabolic equations with nonlinear dynamical boundary conditions (submitted)
Acknowledgments
This work was done while V. Barbu was visiting the Mathematics Department of University of Trento.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Barbu, V., Bonaccorsi, S. & Tubaro, L. A Stochastic Heat Equation with Nonlinear Dissipation on the Boundary. J Optim Theory Appl 165, 317–343 (2015). https://doi.org/10.1007/s10957-014-0672-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10957-014-0672-x