zbMATH — the first resource for mathematics

Stochastic evolution equations with Wick-polynomial nonlinearities. (English) Zbl 1406.60096
Summary: We study nonlinear parabolic stochastic partial differential equations with Wick-power and Wick-polynomial type nonlinearities set in the framework of white noise analysis. These equations include the stochastic Fujita equation, the stochastic Fisher-KPP equation and the stochastic FitzHugh-Nagumo equation among many others. By implementing the theory of \(C_0\)-semigroups and evolution systems into the chaos expansion theory in infinite dimensional spaces, we prove existence and uniqueness of solutions for this class of SPDEs. In particular, we also treat the linear nonautonomous case and provide several applications featured as stochastic reaction-diffusion equations that arise in biology, medicine and physics.

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60H40 White noise theory
60G20 Generalized stochastic processes
37L55 Infinite-dimensional random dynamical systems; stochastic equations
47J35 Nonlinear evolution equations
11B83 Special sequences and polynomials
Full Text: DOI Euclid
[1] Albeverio, S., Di Persio, L. S.: Some stochastic dynamical models in neurobiology: Recent developments. Europena Communications in Mathematical and Theoretical Biology14, (2011), 44–53.
[2] Aronson, D., Weinberger, H.: Nonlinear diffusion in population genetics, combustion and nerve propagation. In J. A. Goldstein, editor, Partial Differential Equations and Related Topics, number 466 in Lecture Notes in Mathematics. Springer–Verlag, New York, 1975. · Zbl 0325.35050
[3] Barbu, V., Cordoni, F., Di Persio, L.S.: Optimal control of stochastic FitzHugh–Nagumo equation. International Journal of Control89(4), (2016), 746–756. · Zbl 1338.93398
[4] Fisher, R.A.: The wave of advance of advantageous genes. Ann. Eug.7, (1937), 353–369. · JFM 63.1111.04
[5] FitzHugh, R.: Impulses and physiological states in theoretical models of nerve membrane. Biophysical Journal1, (1961), 445–466.
[6] Fujita, H., Watanabe, S.: On the uniqueness and non-uniqueness of solutions of initial value problems for some quasi-linear parabolic equations. Comm. Pure. Appl. Math21, (1968), 631–652. · Zbl 0157.17603
[7] Fujita, H., Chen, Y. G.: On the set of blow-up points and asymptotic behaviours of blow-up solutions to a semilinear parabolic equation. Analyse mathématique et applications, 181–201, Gauthier–Villars, Montrouge, 1988.
[8] Hida, T., Kuo, H.-H., Pothoff, J., Streit, L.: White Noise. An Infinite-dimensional Calculus. Kluwer Academic Publishers Group, Dordrecht, 1993.
[9] Holden, H., Øksendal, B., Ubøe, J., Zhang, T.: Stochastic Partial Differential Equations. A Modeling, White Noise Functional Approach. Second Edition. Springer, New York, 2010. · Zbl 1198.60005
[10] Huang, Z., Liu, Z.: Stochastic travelling wave solution to stochastic generalized KPP equation. Nonlinear Differ. Equ. Appl.22, (2015), 143–173. · Zbl 1322.60114
[11] Kaligotla, S.; Lototsky, S. V.: Wick product in the stochastic Burgers equation: a curse or a cure? Asymptot. Anal.75(3-4), (2011), 145–168. · Zbl 1251.35198
[12] Kato, T.: Linear evolution equations of “hyperbolic” type. II. J. Math. Soc. Japan25, (1973), 648–666. · Zbl 0262.34048
[13] Kolmogorov, A., Petrovskii, I., Piskunov, N.: Study of the diffusion equation with increase in the amount of substance and its application to a biological problem. Bull. State Univ. Mos.1, (1937), 1–25.
[14] Levajković, T., Pilipović, S., Seleši, D., Žigić, M.: Stochastic evolution equations with multiplicative noise. Electron. J. Probab.20(19), (2015), 1–23. · Zbl 1321.60137
[15] Meneses, R., Quaas, A.: Fujita type exponent for fully nonlinear parabolic equations and existence results, Journal of Mathematical Analysis and Applications376(2), (2011), 514–527. · Zbl 1218.35027
[16] Mikulevicius, R., Rozovskii, B.: On unbiased stochastic Navier-Stokes equations. Probab. Theory Related Fields154, (2012), 787–834. · Zbl 1277.60109
[17] Nagumo, J., Arimoto, S., Yoshizawa, S.: An active pulse transmission line simulating nerve axon. Proceedings of the Institute of Radio Engineers50(10), (1962), 2061–2070.
[18] Neidhardt, H., Zagrebnov, V. A.: Linear non-autonomous Cauchy problems and evaluation semigroups. Advan. Diff. Equat.14, (2009), 289–340. · Zbl 1180.35371
[19] Øksendal, B., Våge, G., Zhao, H. Z.: Asymptotic properties of the solutions to stochastic KPP equations. Proc. Roy. Soc. Edinburgh Sect. A130(6), (2000), 1363–1381. · Zbl 0978.60070
[20] Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences 44, Springer–Verlag, New York, 1983. · Zbl 0516.47023
[21] Pilipović, S., Seleši, D.: Expansion theorems for generalized random processes, Wick products and applications to stochastic differential equations. Infin. Dimens. Anal. Quantum Probab. Relat. Top.10(1), (2007), 79–110.
[22] Pilipović, S., Seleši, D.: On the generalized stochastic Dirichlet problem - Part I: The stochastic weak maximum principle. Potential Anal.32, (2010), 363–387.
[23] Stanley, R.P.: Catalan Numbers. Cambridge University Press, New York, 2015. · Zbl 1317.05010
[24] Yosida, K.: Time dependent evolution equations in a locally convex space. Math. Ann.162, (1965/1966), 83–86. · Zbl 0138.08301
[25] Zeidler, E.: Nonlinear functional analysis and its applications. I. Fixed-point theorems. Springer-Verlag, New York, 1986. · Zbl 0583.47050
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.