Diffusion equation with a general polynomial perturbation (Q1386264)

Using the renormalization group method [see \textit{J. Bricmont, A. Kupiainen,} and \textit{G. Lin}, Commun. Pure. Appl. Math. 47, 893-922 (1994; Zbl 0806.35067)] the author studies the nonlinear perturbations of the Cauchy problem \[ u_t = \triangle u + F_p(u)\quad (t > 1,\;x \in \mathbb{R}^N),\quad u(x,1) = f(x)\quad (x \in \mathbb{R}^N) \] with small initial data \(f\), where the perturbation \(F_p\) belongs to a quite general class of polynomial operators. (It contains certain polynomial partial differential operators with variable coefficients and also nonlinear integral operators of the type \(F_p(f)(x) = \int _{\mathbb{R}^N} K(x-y) \prod _{j=1}^m D^{n_j} f(y) dy\).) Conditions on \(F_p\) are established under which the solution of the perturbed Cauchy problem behaves asymptotically as the solution of the unperturbed one. Applications of the theory are given.