Existence of global solutions to a nonlinear reaction–fractional diffusion system with anomalous diffusion

In this work, we consider the following fractional reaction system:

, in

, in

or for all , on

for all , in

where u = (u₁, . . . , u_m) , m ≥ 2, Ω is a bounded and regular domain of R^N with boundary ∂Ω, N ≥ 2, u_i = u_i (t, x), 1 ≤ i ≤ m for (t, x) ∈ Q_T = (0, T ) × Ω and ƒ_i are real functions, the presence of the non-local operator , 0<<1 for all 1 ≤ i ≤ m, which accounts for the anomalous diffusion, meaning that the sub-populations face some obstacles that slow their movement, and the constants of diffusion d_i are assumed to be non-negative. ƒ_i : R^m →R^m are regular enough and are non-negative functions in L¹ (Ω) for all cases where 1 ≤ i ≤ m.

The local existence in time of the solution is classical. The positivity of the solution stems from the positivity of , which is assumed to be continuous for all cases where 1 ≤ i ≤ m.