2021/12/21 by Hui Zhan, Fei Gao, Zhan, Hui +3 · 1 citation
Mathematics · #26A33(primary) #35Q92 #35R11(secondary) #92B05 #Analysis of PDEs (math.AP) #Differential Equations and Numerical Methods #FOS: Mathematics #Fractional Differential Equations Solutions #G.1.8 #Nonlinear Differential Equations Analysis #acm:35Q92 #acm:92B05 #math.AP #msc:35Q92 #msc:92B05
paper · pdf · doi:10.48550/arxiv.2112.11143
45 pages
arxiv created 2021/12/21 · openalex publication_date 2021/12/21 · arxiv updated 2021/12/22 · openalex created_date 2022/05/05 · openalex updated_date 2026/07/28
The global boundedness and asymptotic behavior are investigated for the solutions of a nonlocal time fractional reaction-diffusion equation (NTFRDE) \frac∂αu∂ tα=Δu+μu2(1-kJ*u)-γu, (x,t)∈ℝN×(0,+∞) with 0<α<1,β, μ,k>0,N≤ 2 and u(x,0)=u0(x). Under appropriate assumptions on J and the property of time fractional derivative, it is proved that for any nonnegative and bounded initial conditions, the problem has a global bounded classical solution if k*=0 for N=1 or k*=(μC2GN+1)η-1 for N=2, where CGN is the constant in Gagliardo-Nirenberg inequality. With further assumptions on the initial datum, for small μ values, the solution is shown to converge to 0 exponentially or locally uniformly as t → ∞, which is referred as the Allee effect in sense of Caputo derivative. Moreover, under the condition of J ≡ 1, it is proved that the nonlinear NTFRDE has a global bounded solution in any dimensional space with the nonlinear diffusion terms Δum (2-(2)/(N)< m≤ 3).