$L_p$-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 $\dot{F}$: $$ \partial_t u = a^{ij}u_{x^ix^j} + b^i u_{x^i} + cu - \bar{b} u^{1+\beta} + \xi u^{1+\gamma}\dot F,\quad (t,x)\in \mathbb{R}_+\times\mathbb{R}^d; \quad u(0,\cdot) = u_0, $$ 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 H\"older regularity of the solution in time and space.