Abstract :
We consider the Cauchy problem for the damped wave equation
utt − u+ ut = |u|ρ−1u, (t, x) ∈ R+ ×RN
and the heat equation
φt − φ = |φ|ρ−1φ, (t,x) ∈ R+ ×RN.
If the data is small and slowly decays likely c1(1 + |x|)−kN, 0 < k 1, then the critical exponent is ρc(k) = 1 + 2
kN for the
semilinear heat equation. In this paper it is shown that in the supercritical case there exists a unique time global solution to the
Cauchy problem for the semilinear heat equation in any dimensional space RN, whose asymptotic profile is given by
Φ0(t, x) = RN
e−|x−y|2
4t
(4πt)N/2
c1
(1 + |y|2)kN/2 dy
provided that the data φ0 satisfies lim|x|→∞ x kNφ0(x) = c1 ( = 0). Even in the semilinear damped wave equation in the supercritical
case a time global solution u with the data (u, ut )(0, x) = (u0,u1)(x) is shown in low dimensional spaces RN, N = 1, 2, 3,
to have the same asymptotic profile Φ0(t, x) provided that lim|x|→∞ x kN (u0 + u1)(x) = c1 ( = 0). Those proofs are given by
elementary estimates on the explicit formulas of solutions.
© 2007 Elsevier Inc. All rights reserved.
