Global boundedness, hair trigger effect, and pattern formation driven by the parametrization of a nonlocal Fisher-KPP problem

[摘要] The global boundedness and the hair trigger effect of solutions for the nonlinear nonlocal reaction-diffusion equation partial derivative u partial derivative t = Delta u + mu u(alpha)(1-kappa J * u(beta)), in R-N x (0, infinity), N >= 1 with alpha >= 1, beta, mu, kappa > 0 and u(x, 0) = u(0)(x) are investigated. Under appropriate assumptions on J, it is proved that for any nonnegative and bounded initial condition, if alpha is an element of [1, alpha*) with alpha* = 1 + beta for N = 1, 2 and alpha* = 1 + 2 beta/N for N > 2, then the problem has a global bounded classical solution. Under further assumptions on the initial datum, the solutions satisfying 0 <= u(x, t) <= kappa (-1/beta) for any (x, t) is an element of R-N x [0, +infinity) are shown to converge to kappa (-1/beta) uniformly on any compact subset of RN, which is known as the hair trigger effect. 1D numerical simulations of the above nonlocal reaction-diffusion equation are performed and the effect of several combinations of parameters and convolution kernels on the solution behavior is investigated. The results motivate a discussion about some conjectures arising from this model and further issues to be studied in this context. A formal deduction of the model from a mesoscopic formulation is provided as well. (C) 2020 Elsevier Inc. All rights reserved.

[发布日期] 2020-11-15

[关键词] Global boundedness;Hair trigger effect;Pattern formation;Nonlocal reaction-diffusion equation