In this paper, we provide a thorough investigation of the blowing up behavior induced via diffusion of the solution of the following non-local problem: ∂tu = Δu − u + up ∫Ωurdrγ inΩ × (0,T),∂u ∂ν = 0onΓ = ∂Ω × (0,T),u(0) = u0, where Ω is a bounded domain in ℝN with smooth boundary ∂Ω; such problem is derived as the shadow limit of a singular Gierer–Meinhardt system, Kavallaris and Suzuki [On the dynamics of a non-local parabolic equation arising from the Gierer–Meinhardt system, Nonlinearity (2017) 1734–1761; Non-Local Partial Differential Equations for Engineering and Biology: Mathematical Modeling and Analysis, Mathematics for Industry, Vol. 31 (Springer, 2018)]. Under the Turing type condition r p − 1 < N 2 ,γr≠p − 1,p > 1, we construct a solution which blows up in finite time and only at an interior point x0 of Ω, i.e. u(x0,t) ∼ (𝜃∗)− 1 p−1κ(T − t)−1 p−1, where 𝜃∗ :=lim t→T -∫Ωurdr−γandκ = (p − 1)− 1 p−1. More precisely, we also give a description on the final asymptotic profile at the blowup point u(x,T) ∼ (𝜃∗)− 1 p−1 (p − 1)2 8p |x − x0|2 |ln |x − x0||− 1 p−1as x → 0, and thus we unveil the form of the Turing patterns occurring in that case due to driven-diffusion instability. The applied technique for the construction of the preceding blowing up solution mainly relies on the approach developed in [F. Merle and H. Zaag, Reconnection of vortex with the boundary and finite time quenching, Nonlinearity 10 (1997) 1497–1550] and [G. K. Duong and H. Zaag, Profile of a touch-down solution to a nonlocal MEMS model, Math. Models Methods Appl. Sci. 29 (2019) 1279–1348].

中文翻译：

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奇异 Gierer-Meinhardt 系统阴影极限模型的扩散引起的爆破解

在本文中，我们对以下非局部问题的解扩散引起的爆炸行为进行了深入研究： ∂吨你 = Δ你 - 你 + 你p ∫Ω你rdrγ 在Ω × (0,吨),∂你 ∂ν = 0在Γ = ∂Ω × (0,吨),你(0) = 你0, 在哪里Ω是一个有界域ℝñ边界平滑∂Ω;这样的问题被导出为奇异 Gierer-Meinhardt 系统的阴影极限，Kavallaris 和 Suzuki [关于 Gierer-Meinhardt 系统产生的非局部抛物方程的动力学，非线性(2017) 1734–1761；工程和生物学的非局部偏微分方程：数学建模和分析，工业数学，卷。31（斯普林格，2018 年）]。图灵型条件下 r p - 1 < ñ 2 ,γr≠p - 1,p > 1, 我们构建了一个解决方案，它在有限时间内仅在内部点爆炸X0的Ω,IE 你(X0,吨) ～ (𝜃*)- 1 p-1κ(吨 - 吨)-1 p-1, 在哪里 𝜃* ：=林 吨→吨 -∫Ω你rdr-γ和κ = (p - 1)- 1 p-1. 更准确地说，我们还描述了爆炸点处的最终渐近轮廓 你(X,吨) ～ (𝜃*)- 1 p-1 (p - 1)2 8p |X - X0|2 |ln |X - X0||- 1 p-1作为 X → 0, 因此，我们揭示了由于驱动扩散不稳定性而在这种情况下发生的图灵模式的形式。构建上述爆破解决方案的应用技术主要依赖于 [F. Merle 和 H. Zaag，涡旋与边界的重新连接和有限时间淬火，非线性 10(1997) 1497–1550] 和 [GK Duong 和 H. Zaag，非局部 MEMS 模型的触地解决方案简介，数学。模型方法应用程序。科学。 29(2019) 1279–1348]。