$L_p$-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 $\dot{F}$: $${\partial }_{t}u=a^{ij}{u}_{x^i{x}^j}+b^i{u}_{x^i}+cu-\bar{b}{u}^{1+\beta }+\xi u^{1+\gamma }\dot{F},(t,x)\in {\mathbb{R}}_{+}\times {\mathbb{R}}^d;u(0,\cdot )=u_0,$$ where $a^{ij},b^i,c,\bar{b}$ and $\xi$ are $C^2$ or $L_{\infty }$ bounded random coefficients. Here $\beta >0$ denotes the degree of the strong dissipativity and $\gamma >0$ represents the degree of stochastic force. Under the reinforced Dalang's condition on $\dot{F}$, we show the well-posedness of the SRDE provided $\gamma <\frac{\kappa (\beta +1)}{d+2}$ where $\kappa >0$ is the constant related to $\dot{F}$. 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.

21 pages