Asymptotics for nonlinear heat equations

作者：Hayashi Nakao; Naumkin Pavel I^*

来源：Nonlinear Analysis-Theory Methods & Applications, 2011, 74(5): 1585-1595.

DOI：10.1016/j.na.2010.10.029

• Asymptotics of solutions
• Nonlinear heat equations

We study the Cauchy problem for the nonlinear heat equation.
{u(t) - Delta u + u(1+sigma) = 0, x is an element of R-n, t > 0,
u (0, x) = u(0) (x), x is an element of R-n,
in the sub critical case of sigma is an element of (0, 2/n). In the present paper we intend to give a more precise estimate for the remainder term in the asymptotic representation known from paper Escobedo and Kavian (1987) [5]
u(t, x) = t(-1/sigma) w(0) (x/root t) + o(t(-1/sigma))
as t -> infinity uniformly with respect to x is an element of R-n, where w(0) (xi) is a positive solution of equation
-Delta w - xi/2 . del w + w(1+sigma) = 1/sigma w
which decays rapidly at infinity: lim(vertical bar xi vertical bar ->+/-infinity) vertical bar xi vertical bar(2/sigma) w(0) (xi) = 0.

• 出版日期2011-3-1