Large Time Behavior of Solutions to Semilinear Parabolic Equations with Gradient

In this paper, we investigate the large time behavior of solutions to the Cauchy problem of a class of semilinear parabolic equations with gradient. The blowing-up theorem of Fujita type is established, and the critical Fujita exponent is formulated by the spacial dimension and the behavior of the coefficient of the gradient term at infinity. It is a surprise that the critical Fujita exponent even could be 1 or infinite due to the effect of the gradient term. The critical case is also considered. An interesting phenomenon is that the critical Fujita exponent can belong to not only the blowing-up case but also the global existence case for each nontrivial solution.