Liouville Theorems for Semilinear Parabolic Equations Involving the Nonlinear Gradient Term

In this paper, we prove the Liouville-type theorem for equation $u_t-\mathrm{\Delta }u=u^p+M\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^q$ via the integral Bernstein method, where $q=2p\mathrm{/}(p+1)$ is critical and $M<0$. As an application, we can establish a universal priori estimate for this equation by using the doubling lemma. These results generalize the previous work of Liang and Zhang. We can also prove the Liouville-type theorem for equation $u_t-\mathrm{\Delta }u=u^p+M\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^2$, which corresponds to the supercritical case.