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 -