Analysis on Fisher-KPP type nonlocal reaction diffusion equations
Final Report Abstract
Reaction diffusion equations are well known models in population dynamics and biomedical processes, which have been well studied mathematically before the 1980s. However, the results are not valid for the newly introduced diffusion equation with nonlocal reactions. On the other hand, the nonlocal reaction terms describe the emergence and propagation of spatial structures which do not exist in the local case. The analysis on diffusion equations with nonlocal reaction effects is a challenging research area in PDE analysis. The main nonlocal reaction terms considered in this project is of nonlocal Fisher-KPP (Kolmogorov, Petrovskii, Piskunov) type. The global existence of bounded solutions, in either bounded domain or whole space domain, have been obtained both for diffusion equation and for the chemotaxis system with the same nonlocal nonlinear reaction effects in the subcritical exponent. An important effect arises from the results is that the critical exponent appeared from nonlocal nonlinear reaction is exactly the Fujita exponent in nonlinear reaction, which reveal the important effect from nonlocal reaction. Within the subcritical region, according to the Fujita theory the solution will blow up in finite time, while by switching on the nonlocal reaction one can obtain a global bounded solution. The result in chemotaxis model shows that nonlocal nonlinear reaction term can help in preventing the blow up from aggregation effect. Another independent result obtained in this project is the travelling wave solution in the double stationary case. One can see the rich behaviors of the solutions that are not observed in the local case. It is obtained that there exists a critical speed, such that below this speed, a monotone wavefront can be connected by the two positive equilibrium points. On the other hand, there exists another speed threshold such that the model admits a semi-wavefront. When the convolution potential is approaching to Dirac delta function, the semi-wavefronts are in fact wavefronts connecting 0 to the largest equilibrium. In addition, the wavefronts converge to those of the local problem. There are much richer phenomena one can observe with the nonlocal reaction terms and more theories, for example in the super critical region whether the solution will be bounded or blow up, more detailed observation on the wavefront solutions when the wave speed exceed the threshold or the convolution potential is relatively wide, need to be developed in the future.
Publications
- A nonlocal reaction diffusion equation and its relation with Fujita exponent, Journal of Mathematical Analysis and Applications 444 (2), 1479-1489, 2016
S Bian, L Chen
(See online at https://doi.org/10.1016/j.jmaa.2016.07.014) - Global existence and asymptotic behavior of solutions to a nonlocal Fisher–KPP type problem, Nonlinear Analysis: Theory, Methods & Applications 149, 165-176, 2017
S Bian, L Chen, EA Latos
(See online at https://doi.org/10.1016/j.na.2016.10.017) - Wavefronts for a nonlinear nonlocal bistable reaction–diffusion equation in population dynamics, Journal of Differential Equations 263 (10), 6427-6455, 2017
J Li, E Latos, L Chen
(See online at https://doi.org/10.1016/j.jde.2017.07.019) - Chemotaxis model with nonlocal nonlinear reaction in the whole space, Discrete & Continuous Dynamical Systems-A 38 (10), 5067- 5083, 2018
S Bian, L Chen, EA Latos
(See online at https://doi.org/10.3934/dcds.2018222) - Nonlocal nonlinear reaction preventing blow-up in supercritical case of chemotaxis system, Nonlinear Analysis 176, 178-191, 2018
S Bian, L Chen, EA Latos
(See online at https://doi.org/10.1016/j.na.2018.06.012)
