Abstract
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.
Highlights
IntroductionThe present paper is an extension of [14] to nonlinear equations, where the nonlinearity is generated by a Wick-polynomial function leading to stochastic versions of Fujita-type equations ut = Au + u♦n + f , FitzHugh-Nagumo equations ut = Au + u♦2 − u♦3 + f , Fisher-KPP equations ut = Au + u − u♦2 + f and Chaffee-Infante equations ut = Au + u♦3 − u + f
We study stochastic nonlinear evolution equations of the form n ut(t, ω) = A u(t, ω) + aku♦k(t, ω) + f (t, ω), k=0 u(0, ω) = u0(ω), ω ∈ Ω, t ∈
The present paper is an extension of [14] to nonlinear equations, where the nonlinearity is generated by a Wick-polynomial function leading to stochastic versions of Fujita-type equations ut = Au + u♦n + f, FitzHugh-Nagumo equations ut = Au + u♦2 − u♦3 + f, Fisher-KPP equations ut = Au + u − u♦2 + f and Chaffee-Infante equations ut = Au + u♦3 − u + f
Summary
The present paper is an extension of [14] to nonlinear equations, where the nonlinearity is generated by a Wick-polynomial function leading to stochastic versions of Fujita-type equations ut = Au + u♦n + f , FitzHugh-Nagumo equations ut = Au + u♦2 − u♦3 + f , Fisher-KPP equations ut = Au + u − u♦2 + f and Chaffee-Infante equations ut = Au + u♦3 − u + f. These equations have found ample applications in ecology, medicine, engineering and physics.
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