Abstract
We investigate stochastic reaction-diffusion equations on finite metric graphs. On each edge in the graph a multiplicative cylindrical Gaussian noise driven reaction-diffusion equation is given. The vertex conditions are the standard continuity and generalized, non-local Neumann-Kirchhoff-type law in each vertex. The reaction term on each edge is assumed to be an odd degree polynomial, not necessarily of the same degree on each edge, with possibly stochastic coefficients and negative leading term. The model is a generalization of the problem in [14] where polynomials with much more restrictive assumptions are considered and no first order differential operator is involved. We utilize the semigroup approach from [15] to obtain existence and uniqueness of solutions with sample paths in the space of continuous functions on the graph.
Article PDF
Similar content being viewed by others
References
F. Albiac and N. J. Kalton, Topics in Banach Space Theory, 2nd ed., Graduate Texts in Mathematics, vol. 233, Springer (Cham, 2016) (with a foreword by Gilles Godefroy).
W. Arendt, Semigroups and evolution equations: functional calculus, regularity and kernel estimates, in: Evolutionary Equations, vol. I, Handb. Differ. Equ., North-Holland (Amsterdam, 2004), pp. 1–85.
W. Arendt, C. J. K. Batty, M. Hieber, and F. Neubrander, Vector-valued Laplace Transforms and Cauchy Problems, 2nd ed., Monographs in Mathematics,vol. 96, Birkhäuser/Springer Basel AG (Basel, 2011).
G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs, Mathematical Surveys and Monographs, vol. 186, American Mathematical Society (Providence, RI, 2013).
S. Bonaccorsi, C. Marinelli, and G. Ziglio, Stochastic FitzHugh–Nagumo equations on networks with impulsive noise, Electron. J. Probab., 13 (2008), 1362–1379.
S. Bonaccorsi and G. Ziglio, Existence and stability of square-mean almost periodic solutions to a spatially extended neural network with impulsive noise, Random Oper. Stoch. Equ., 22 (2014), 17–29.
S. Cerrai, Stochastic reaction-diffusion systems with multiplicative noise and non- Lipschitz reaction term, Probab. Theory Related Fields, 125 (2003), 271–304.
F. Cordoni and L. Di Persio, Gaussian estimates on networks with dynamic stochastic boundary conditions, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 20 (2017), 1750001, 23 pp.
F. Cordoni and L. Di Persio, Stochastic reaction-diffusion equations on networks with dynamic time-delayed boundary conditions, J. Math. Anal. Appl., 451 (2017),583–603.
L. Di Persio and G. Ziglio, Gaussian estimates on networks with applications to optimal control, Netw. Heterog. Media, 6 (2011), 279–296.
K.-J. Engel and R. Nagel,One-parameter Semigroups for Linear Evolution Equations,,Graduate Texts in Mathematics, vol. 194, Springer-Verlag (New York, 2000) (with contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt).
M. Haase, The Functional Calculus for Sectorial Operators, Operator Theory: Advances and Applications, vol. 169, Birkhäuser Verlag (Basel, 2006).
M. Kovács and E. Sikolya, On the stochastic Allen–Cahn equation on networks with multiplicative noise, Electron. J. Qual. Theory Differ. Equ. (2021), Paper No. 7,24 pp.
M. Kovács and E. Sikolya, Corrigendum to “On the stochastic Allen–Cahn equation on networks with multiplicative noise”, Electron. J. Qual. Theory Differ. Equ.(2021), Paper No. 52, 4 pp.
M.Kovács and E. Sikolya, Stochastic reaction-diffusion equations on networks, J. Evol. Equ., 21 (2021), 4213–4260.
M. Kunze and J. van Neerven, Continuous dependence on the coefficients and global existence for stochastic reaction diffusion equations, J. Differential Equations,253 (2012), 1036–1068.
M. Kunze and J. van Neerven, Continuous dependence on the coefficients and global existence for stochastic reaction diffusion equations, arXiv: 1104.4258v3 (2019).
G. A. Muñoz, Y. Sarantopoulos and A. Tonge, Complexifications of real Banach spaces, polynomials and multilinear maps, Studia Math., 134 (1999), 1–33.
D. Mugnolo, Gaussian estimates for a heat equation on a network, Netw. Heterog. Media, 2 (2007), 55–79.
D. Mugnolo and S. Romanelli, Dynamic and generalized Wentzell node conditions for network equations, Math. Methods Appl. Sci., 30 (2007), 681–706.
E. M. Ouhabaz, Analysis of Heat Equations on Domains, London Mathematical Society Monographs Series, vol. 30, Princeton University Press (2004).
H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland Mathematical Library, vol. 18, North-Holland Publishing Co.(Amsterdam–New York, 1978).
J. van Neerven, The Adjoint of a Semigroup of Linear Operators, Lecture Notes in Math., vol. 1529, Springer-Verlag (Berlin, 1992).
J. M. A. M. van Neerven, M. C. Veraar and L. Weis, Stochastic evolution equations in UMD Banach spaces, J. Funct. Anal., 255 (2008), 940-993.
Author information
Authors and Affiliations
Corresponding author
Additional information
The author was supported by the OTKA grant no. 135241.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Sikolya, E. Reaction-diffusion equations on metric graphs with edge noise. Anal Math (2024). https://doi.org/10.1007/s10476-024-00006-z
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10476-024-00006-z