Asymptotics of solutions of nonlinear parabolic equations. (Q1399902)

The paper is concerned with the large time behaviour of solutions to the Cauchy problem for the nonlinear parabolic equations \[ u_t = \Delta u + F(u, D_x u, D^2_x u), \qquad u \in \mathbb R^n,\;x \in \mathbb R^N, t > 0 \] with initial data \[ u(0,x) = u_0(x), \qquad x \in \mathbb R^N,\;N \geq 1, \] \(F\) being a sufficiently smooth vector-valued function. Under appriopriate assumptions the following results are obtained: (1) the existence of a unique global smooth solution provided the initial data are sufficiently small in a suitably defined norm; (2) the decay estimates for the solution \(u(t,x)\) and for its derivatives, when \(t \geq \tau >0\). The detailed exposition of earlier results in the considered topic and the references are given.