2019/09/17 by Li, Jing, Chen, Li, Surulescu, Christina · 2 citations
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.1909.07934
The global boundedness and the hair trigger effect of solutions for the nonlinear nonlocal reaction-diffusion equation ut=Δu+μuα(1-κJ*uβ), \hboxin \mathbb RN×(0,∞), N≥ 1 with α≥1, β,μ,κ>0 and u(x,0)=u0(x) are investigated. Under appropriate assumptions on J, it is proved that for any nonnegative and bounded initial condition, if α∈[1,α^*) with α^*=1+β for N=1,2 and α^*=1+(2β)/(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)≤κ-\frac1β for any (x,t)∈\mathbb RN×[0,+∞) are shown to converge to κ-\frac1β uniformly on any compact subset of \mathbb 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.