# The explicit solution to the initial–boundary value problem of Gierer–Meinhardt model

Research paper by Xiaowei An, Zhen He; Xianfa Song

Indexed on: 02 Mar '18Published on: 05 Feb '18Published in: Applied Mathematics Letters

#### Abstract

Publication date: June 2018 Source:Applied Mathematics Letters, Volume 80 Author(s): Xiaowei An, Zhen He, Xianfa Song Using the solutions of an elliptic system and an ODE system, under certain conditions, we get the explicit solution to the following initial–boundary value problem of Gierer-Meinhardt model u t = d 1 Δ u − a 1 u + u p v q + δ 1 ( x , t ) , x ∈ Ω , t > 0 v t = d 2 Δ v − a 2 v + u r v s + δ 2 ( x , t ) , x ∈ Ω , t > 0 ∂ u ∂ η = ∂ v ∂ η = 0 ( or u = v = 0 ) , x ∈ ∂ Ω , t > 0 u ( x , 0 ) = u 0 ( x ) , v ( x , 0 ) = v 0 ( x ) , x ∈ Ω . Here p > 1 , s > − 1 , d 1 , d 2 , q , r > 0 and a 1 , a 2 ≥ 0 are constants, while δ 1 ( x , t ) and δ 2 ( x , t ) are nonnegative continuous functions.