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We study the existence, uniqueness, and regularity of the solution to thestochastic reaction-diffusion equation (SRDE) with colored noise $\dot{F}$: $$\partial_t u = a^{ij}u_{x^ix^j} + b^i u_{x^i} + cu - \bar{b} u^{1+\beta} + \xiu^{1+\gamma}\dot F,\quad (t,x)\in \mathbb{R}_+\times\mathbb{R}^d; \quadu(0,\cdot) = u_0, $$ 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 ofthe strong dissipativity and $\gamma>0$ represents the degree of stochasticforce. Under the reinforced Dalang's condition on $\dot{F}$, we show thewell-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 thatstrong dissipativity prevents the solution from blowing up. Moreover, weprovide the maximal H\"older regularity of the solution in time and space.

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