Lp-regularity theory for the stochastic reaction-diffusion equation with super-linear multiplicative noise and strong dissipativity

Abstract

We study the existence, uniqueness, and regularity of the solution to the stochastic reaction-diffusion equation (SRDE) with colored noise F˙:∂tu=aijuxixj+biuxi+cu−b¯u1+β+ξu1+γF˙,(t,x)∈R+×Rd;u(0,⋅)=u0, where aij,bi,c,b¯ and ξ are C2 or L∞ bounded random coefficients. Here β>0 denotes the degree of strong dissipativity and γ>0 represents the degree of stochastic force. Under the reinforced Dalang's condition on F˙, we show the well-posedness of the SRDE provided γ<κ(β+1)d+2 where κ>0 is the constant related to F˙. Our result assures that strong dissipativity prevents the solution from blowing up. Moreover, we provide the maximal Hölder regularity of the solution in time and space.