Positive probability of explosion for stochastic heat equation with superlinear accretive reaction term and polynomially growing multiplicative noise

This paper studies the finite time explosion of the stochastic heat equation $\frac{\mathrm{\partial }u}{\mathrm{\partial }t}(t,x)=\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}^2}u(t,x)+(u(t,x){)}^{\beta }+\sigma (u(t,x))\dot{W}(t,x)$. We consider an interval $D=[-\pi ,\pi ]$ under periodic boundary condition where $\dot{W}(t,x)$ is a space-time white noise and $\sigma (u)\approx u^{\gamma }$ near $\mathrm{\infty }$. Our results refine existing results by identifying behavior in a previously less understood regime, where we show that if $\beta \in (1,3),\gamma \in (\frac{\beta }{2},\frac{\beta +3}{4})$ or $\beta >1,\gamma \in (0,\frac{\beta }{2}]$ then mild solutions can explode with positive probability. This paper provides a partial characterization of the explosion behavior in an intermediate parameter regime, and contribute to the understanding of the interplay between the drift and diffusion terms.