Properties Of Solutions To A Nonlocal Diffusion Problem With Localized Nonlinear Reaction Terms

Properties of solutions to nonlinear partial differential equations are always one of the most popular topics in the research of nonlinear analysis and partial differential equations, and these equations also have close relationships to many mathematical models in the applied disciplines, such as biology, chemistry and physics. With the rapid development of science and technology and with the improvement of mathematical methods, the forms of nonlinear partial differential equations are more and more diverse.In recent years, more and more scientific workers have paid their attentions to nonlocal diffusion equation of this kind This model has been used to describe the process of diffusion, where u can be interpreted as the density of a single population at point x at time t, and J(x-y) as the probability distribution of jumping from location y to location x. Then, the convolution ly is the rate at which they are arriving to location x from all other places.This paper mainly consider solutions to the following nonlocal reaction-diffusion system with nonlinear reaction termsWe are to discuss properties such as existence, uniqueness, and long-time behavior, etc. In particular, when blowup in finite time happens, properties related to blowup are to be further discussed:some sufficient conditions and some necessary conditions for simultaneous blowup and non-simultaneous blowup of u and v, and blow-up profile of solutions (u, v).We firstly apply the Contraction Mapping Principle to get existence and uniqueness of the solution; Secondly, by establishing a new comparison principle and by constructing suitable sup-solutions and sub-solutions, conditions for finite time blowup of solutions are also concluded; And then, blowup profile of solutions (u,v) is also obtained mainly by virtue of inequalities, the ordinary differential inequalities and the comparison principle; Finally, due to the comparison principle and series of general analytical methods, by carefully deducing we arrive at some sufficient conditions and some necessary conditions for simultaneous blowup and non-simultaneous blowup of u and v. Furthermore, estimates of blow-up rate and blow-up set are also established.