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Asymptotics of solutions of nonlinear parabolic equations. (English) Zbl 1036.35064
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.

MSC:
35K15 Initial value problems for second-order parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
35K55 Nonlinear parabolic equations
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