Global solutions and self-similar solutions of semilinear parabolic equations with nonlinear gradient terms

In this paper we consider the Cauchy problem of semilinear parabolic equations with nonlinear gradient terms a(x)|u|q−1u|∇u|pa(x)|u|q−1u|∇u|p. We prove the existence of global solutions and self-similar solutions for small initial data. Moreover, for a class of initial data we show that the global solutions behave asymptotically like self-similar solutions as t→∞t→∞.