Lp-regularity theory for semilinear stochastic partial differential equations with multiplicative white noise

We establish the L-p-regularity theory for a semilinear stochastic partial differential equation with multiplicative white noise: du = (a(ij)u(xixj) + b(i)u(xi) + cu +(b) over bar (i)vertical bar u vertical bar(lambda)u(xi)) dt + sigma(k)(u)dw(t)(k) , (omega, t, x) is an element of Omega x (0 , infinity) x R-d; u(0 , .) = u(0), where lambda > 0, the set {w(t)(k), k = 1 , 2 , ...} is a set of one-dimensional independent Wiener processes, and the function u(0) = u(0)(omega, x) is nonnegative random initial data. The coefficients a(ij), b(i), c depend on (omega, t, x), and (omega, t, x(1) , .. ., x(i-1) , x(i+1) , .. ., x(d)). The coefficients a(ij), b(i), c, <(b)over bar >(i) are uniformly bounded and twice continuously differentiable. The leading coefficient a satisfies the ellipticity condition. Depending on the diffusion coefficient sigma(k)(u) = sigma(k)(omega, t, x, u), we consider two cases: (i) lambda is an element of (0 , infinity) and sigma(k)(u) has a Lipschitz continuity and linear growth in u. (ii) lambda, lambda(0) is an element of (0 , 1/d) and sigma(k)(omega, t, x, u) = mu k(omega, t, x)vertical bar u vertical bar(1+lambda 0) with vertical bar Sigma(k)vertical bar mu(k)(omega, t, .)vertical bar(2)vertical bar(C(Rd)) < infinity for all (omega, t) is an element of Omega x [0, infinity). We obtain the existence, uniqueness, L-p regularity, and Holder regularity of the solution. It should be noted that each case has different regularity results. For example, in the case of (i), for epsilon > 0 u is an element of C-t,x(1/2-epsilon,1-epsilon) ([0 , T] x R-d) for all T < infinity, almost surely. On the other hand, in the case of (ii), if lambda, lambda(0) is an element of (0 , 1/d), for epsilon > 0 u is an element of C-t,x(1-(lambda d)for all(lambda 0d) -epsilon,1-(lambda d)for all(lambda 0d)- 2 epsilon)([0 , T] x R-d) for all T < infinity almost surely. It should be noted that lambda can be any positive number and that the solution regularity is independent of the nonlinear terms in case (i). In case (ii), however, lambda and lambda(0) should satisfy lambda, lambda(0) is an element of (0 , 1/d), and the regularities of the solution are affected by lambda, lambda(0) and d. This difference is due to the need to use a different proof method for the super-linear diffusion coefficient sigma(k)(u). (c) 2022 Elsevier Inc. All rights reserved.