Diffusion-induced blowup solutions for the shadow limit model of a singular Gierer-Meinhardt system

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: {partial derivative(t)u = Delta u - u + u(p)/f(Omega)u(r)dr)(gamma) in Omega x (0, T), partial derivative u/partial derivative nu = 0 on Gamma = partial derivative Omega x (0, T), u(0) = u0, where Omega is a bounded domain in Double-struck capital R-N with smooth boundary partial derivative Omega; 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 ,gamma r not equal p - 1, p > 1, we construct a solution which blows up in finite time and only at an interior point x(0) of Omega, i.e. u(x(0), t) similar to (theta*)-(1/p-1)[kappa(T - t)(-1/p-1)], where theta* := lim(t -> T) (-integral(Omega)u(r)dr)(-gamma) and kappa = (p - 1)(-1/p-1). More precisely, we also give a description on the final asymptotic profile at the blowup point u(x, T) similar to (theta*)(-1/p-1) [(p - 1)(2)/8p vertical bar x - x(0)vertical bar(2)/vertical bar ln vertical bar x - x(0)vertical bar vertical bar](-1/p-1) as 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].

Automated summary

This paper thoroughly investigates the blowing up behavior induced through diffusion of the solution to a non-local problem, providing constructions of solutions that blow up in finite time at interior points under certain conditions. The final asymptotic profile at the blowup point and the form of Turing patterns occurring in that case are also described. The technique used for constructing the blowing up solutions mainly relies on previously developed approaches.