In this work, we consider the stochastic version of the diffusion equations with polynomial reaction terms forced by a multiplicative white noise. We establish the existence and uniqueness of a maximal pathwise solution for a limited period of time. The proof relies on the Skorohod representation theorem, the Gyöngy–Krylov theorem and stopping time arguments.
P.Billingsley, Probability and Measure, 3rd edn, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1995. A Wiley-Interscience Publication.
4.
P.Constantin and C.Foias, Navier–Stokes Equations, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1988.
5.
G.Da Prato and J.Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and Its Applications, Vol. 44, Cambridge University Press, Cambridge, 1992.
6.
A.Debussche, N.Glatt-Holtz and R.Temam, Local martingale and pathwise solutions for an abstract fluids model, Phys. D240(14–15) (2011), 1123–1144. doi:10.1016/j.physd.2011.03.009.
C.R.Doss, M.A.Suchard, I.Holmes, M.Kato-Maeda and V.N.Minin, Fitting birth–death processes to panel data with applications to bacterial DNA fingerprinting, Ann. Appl. Stat.7(4) (2013), 2315–2335. doi:10.1214/13-AOAS673.
9.
N.H.Du and V.H.Sam, Dynamics of a stochastic Lotka–Volterra model perturbed by white noise, J. Math. Anal. Appl.324(1) (2006), 82–97. doi:10.1016/j.jmaa.2005.11.064.
10.
L.C.Evans, Weak Convergence Methods for Nonlinear Partial Differential Equations, CBMS Regional Conference Series in Mathematics, Vol. 74, Washington, DC, 1990. Published for the Conference Board of the Mathematical Sciences. doi:10.1090/cbms/074.
11.
L.C.Evans, An Introduction to Stochastic Differential Equations, American Mathematical Society, Providence, RI, 2013.
12.
F.Flandoli, A stochastic reaction–diffusion equation with multiplicative noise, Appl. Math. Lett.4(4) (1991), 45–48. doi:10.1016/0893-9659(91)90052-W.
13.
F.Flandoli, An introduction to 3D stochastic fluid dynamics, in: SPDE in Hydrodynamic: Recent Progress and Prospects, Lecture Notes in Math., Vol. 1942, Springer, Berlin, 2008, pp. 51–150. doi:10.1007/978-3-540-78493-7_2.
14.
F.Flandoli and D.Gatarek, Martingale and stationary solutions for stochastic Navier–Stokes equations, Probab. Theory Related Fields102(3) (1995), 367–391. doi:10.1007/BF01192467.
15.
G.B.Folland, Real Analysis: Modern Techniques and Their Applications, 2nd edn, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1999. A Wiley-Interscience Publication.
16.
S.-C.Fu and J.-S.Guo, Blow-up for a semilinear reaction–diffusion system coupled in both equations and boundary conditions, J. Math. Anal. Appl.276(1) (2002), 458–475. doi:10.1016/S0022-247X(02)00506-1.
17.
N.Glatt-Holtz and M.Ziane, Strong pathwise solutions of the stochastic Navier–Stokes system, Adv. Differential Equations14(5–6) (2009), 567–600.
18.
I.Gyöngy and N.Krylov, Existence of strong solutions for Itô’s stochastic equations via approximations, Probab. Theory Related Fields105(2) (1996), 143–158. doi:10.1007/BF01203833.
19.
D.Horstmann, Remarks on some Lotka–Volterra type cross-diffusion models, Nonlinear Anal. Real World Appl.8(1) (2007), 90–117. doi:10.1016/j.nonrwa.2005.05.008.
20.
T.Kawamura and Y.Saisho, Stochastic models describing human metabolic processes using SDEs with reflection, Stoch. Models22(2) (2006), 273–287. doi:10.1080/15326340600649037.
21.
J.Link, P.Nguyen and R.Temam, Local solutions to the stochastic one layer shallow water equations, JMAA, accepted.
22.
J.-L.Lions and E.Magenes, Non-homogeneous Boundary Value Problems and Applications, Vol. I, Die Grundlehren der mathematischen Wissenschaften, Vol. 181, Springer-Verlag, New York, 1972. Translated from the French by P. Kenneth.
23.
Y.Masuda, First passage times of birth–death processes and simple random walks, Stochastic Process. Appl.29(1) (1988), 51–63. doi:10.1016/0304-4149(88)90027-0.
24.
L.-S.Meng, P.-C.Yeh, K.-C.Chen and I.F.Akyildiz, On receiver design for diffusion-based molecular communication, IEEE Trans. Signal Process.62(22) (2014), 6032–6044. doi:10.1109/TSP.2014.2359644.
25.
T.Nakano, M.J.Moore, F.Wei, A.V.Vasilakos and J.Shuai, Molecular communication and networking: Opportunities and challenges, IEEE Trans. Nanobioscience11(2) (2012), 135–148. doi:10.1109/TNB.2012.2191570.
26.
K.Nakashima and Y.Yamada, Positive steady states for prey–predator models with cross-diffusion, Adv. Differential Equations1(6) (1996), 1099–1122.
27.
B.Øksendal, Stochastic Differential Equations, 6th edn, Universitext, Springer-Verlag, Berlin, 2003. An introduction with applications.
28.
C.Prévôt and M.Röckner, A Concise Course on Stochastic Partial Differential Equations, Lecture Notes in Mathematics, Vol. 1905, Springer, Berlin, 2007.
29.
A.Puhalskii and B.Simon, Discrete evolutionary birth–death processes and their large population limits, Stoch. Models28(3) (2012), 388–412. doi:10.1080/15326349.2012.699752.
30.
M.R.Sheldon, Stochastic Processes, 2nd edn, Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons, Inc., New York, 1996.
31.
R.Temam, Navier–Stokes Equations and Nonlinear Functional Analysis, 2nd edn, CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 66, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1995.
32.
R.Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2nd edn, Applied Mathematical Sciences, Vol. 68, Springer-Verlag, New York, 1997.
33.
R.Temam, Navier–Stokes Equations, AMS Chelsea Publishing, Providence, RI, 2001. Theory and numerical analysis, Reprint of the 1984 edition.