On asymptotic ``eigenfunctions'' of the Cauchy problem for a~nonlinear parabolic equation

V. A. Galaktionov ; S. P. Kurdyumov ; A. A. Samarskii

Sbornik. Mathematics, Tome 54 (1986) no. 2, pp. 421-455

Résumé

The asymptotic ($t\to+\infty$) behavior of solutions of the Cauchy problem is studied for the semilinear parabolic equation $$ u_t=\Delta u-u^\beta,\quad t>0,\ x\in R^N;\qquad u(0,x)=u_0(x)\geqslant0,\quad x\in R^N, $$ where $\beta=\mathrm{const}>1$ and $u_0(x)\to0$ as $|x|\to+\infty$. The existence is established of an infinite collection (a continuum) of distinct self-similar solutions of the form $u_A(t,x)=(T+t)^{-1/(\beta-1)}\theta_A(\xi)$, $\xi=|x|/(T+t)^{1/2}$, where the function $\theta_A>0$ satisfies an ordinary differential equation. Conditions for the asymptotic stability of these solutions are established. It is shown that for $\beta\geqslant1+2/N$ there exist solutions of the problem whose behavior as $t\to+\infty$ is described by approximate self-similar solutions (ap.s.-s.s.'s) $u_a(t,x)$ which in the case $\beta>1+2/N$ coincide with a family of self-similar solutions of the heat equation $(u_a)_t=\Delta u_a$, while for $\beta=1+2/N$ and $u_0\in L^1(R^N)$ the ap.s.-s.s. has the form $u_a=[(T+t)\ln(T+t)]^{-N/2}c_N\exp(-|x|^2/4(T+t))$, where $c_N=(N/2)^{N/2}(1+2/N)^{N^2/4}$. Figures: 2. Bibliography: 78 titles.