TY - JOUR
T1 - Refined asymptotic profiles for a semilinear heat equation
AU - Ishige, Kazuhiro
AU - Kawakami, Tatsuki
N1 - Funding Information:
K. Ishige was Supported in part by the Grant-in-Aid for Scientific Research (B) (No. 23340035), Japan Society for the Promotion of Science.
PY - 2012/5
Y1 - 2012/5
N2 - We study the large time behavior of the solutions of the Cauchy problem for a semilinear heat equation, ∂ tu = Δu+F(x,t,u) in R N × (0, ∞) u(x, 0) = φ(x) in R N, (P) where F ∈ C(R N × [0, ∞) × R) and φ ∈ L 1 (R N, (1 + {pipe}x{pipe}) K dx) with K ≥ 0. Assume that u is a solution of (P) satisfying {pipe}F(x,t,u(x,t)){pipe} ≤ C(1+t) -A {pipe}u(x,t){pipe}, (x,t) ∈ R N × (0,∞) for some constants C > 0 and A > 1. Then it is well known that the solution u behaves like the heat kernel. In this paper we give the ([K] + 2)th order asymptotic expansion of the solution u, and reveal the relationship between the asymptotic profile of the solution u and the nonlinear term F. Here [K] is the integer satisfying K - 1 < [K] ≤ K.
AB - We study the large time behavior of the solutions of the Cauchy problem for a semilinear heat equation, ∂ tu = Δu+F(x,t,u) in R N × (0, ∞) u(x, 0) = φ(x) in R N, (P) where F ∈ C(R N × [0, ∞) × R) and φ ∈ L 1 (R N, (1 + {pipe}x{pipe}) K dx) with K ≥ 0. Assume that u is a solution of (P) satisfying {pipe}F(x,t,u(x,t)){pipe} ≤ C(1+t) -A {pipe}u(x,t){pipe}, (x,t) ∈ R N × (0,∞) for some constants C > 0 and A > 1. Then it is well known that the solution u behaves like the heat kernel. In this paper we give the ([K] + 2)th order asymptotic expansion of the solution u, and reveal the relationship between the asymptotic profile of the solution u and the nonlinear term F. Here [K] is the integer satisfying K - 1 < [K] ≤ K.
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U2 - 10.1007/s00208-011-0677-9
DO - 10.1007/s00208-011-0677-9
M3 - Article
AN - SCOPUS:84859823058
VL - 353
SP - 161
EP - 192
JO - Mathematische Annalen
JF - Mathematische Annalen
SN - 0025-5831
IS - 1
ER -