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

We study the existence, uniqueness, and regularity of the solution to the stochastic reaction-diffusion equation (SRDE) with colored noise F-center dot:partial derivative(t)u = aijuxixj +biuxi + cu - bu1+beta +xi u1+gamma F-center dot, (t, x) is an element of R+ x Rd; u(0, ) = u0,where a(ij), b(i), c, b and xi are C-2 or L-infinity bounded random coefficients. Here beta > 0 denotes the degree of strong dissipativity and gamma > 0 represents the degree of stochastic force. Under the reinforced Dalang's condition on F-center dot, we show the well-posedness of the SRDE provided gamma < kappa (beta+1) d+2 where kappa > 0 is the constant related to F-center dot. Our result assures that strong dissipativity prevents the solution from blowing up. Moreover, we provide the maximal H & ouml;lder regularity of the solution in time and space. (c) 2023 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons .org /licenses /by /4 .0/).