The Properties Of Solutions For Nonlocal Reaction-Diffusion Equations And Systems With Weighted Term

As one of the most important partial differential equations and systems to describe diffusion phenomenon,the reaction-diffusion equations and systems are widely concerned and studied by researchers in various disciplines.Among them,the study on blowup behavior of the solution for this kind of equations and systems has became an important research branch,because it can predict the blowup time well.over the past decades,many important findings have been achieved through the efforts of domestic and foreign scholars.For the lateliest years,with the continuous deepening and development of research contents and methods,many researchers turn attention to a class of reaction-diffusion equations and systems,which diffusion terms are represented by convolution operators and reaction terms by integral.Considering its extensive practical application backgrounds and theoretical value,based on the basic knowledge of partial differential equations and integral theory,this thesis studies the properties of solutions for several nonlocal diffusion equations and systems with weighted reaction terms,including the local existence and blowup of solutions.This thesis mainly discusses the sufficient conditions to be satisfied on blowup,and further estimates the bounds of blowup time.Firstly,the actual backgrounds and development process of reaction-diffusion equations and systems are introduced systematically,and some important achievements made by researchers at home and abroad in the past decades are summarized.Influenced and inspired by these papers,this thesis discusses the blowup properties of solutions for the nonlocal reaction-diffusion equations and systems,which have attracted much attention in recent years.Secondly,the properties of solutions for the time-weighted nonlocal diffusion equation under the first boundary condition are investigated.The local existence of solutions is verified by using the compression mapping principle in L~?((?)) space.Then by using the method of characteristic function and combining with a series of inequalities,the sufficient conditions for the solution of blowup are explored.Further the upper bound of blowup time is estimated by differential and integral methods.Thirdly,under the first boundary condition,the properties of solutions for the time-weighted nonlocal diffusion systems are studied.By introducing abstract semigroup theory and applying Bananch fixed point theorem,the local existence of solutions in intersection space L~1((?))?L~?((?)) is proved.In addition,by making some appropriate assumptions about the reaction terms,and constructing a new auxiliary function,finds the sufficient conditions for the solution of blowup,the upper bound of blowup time is derived.Finally,consider the properties of solutions for the position-weighted nonlocal diffusion systems under the first boundary condition.The Bananch fixed point theory is used to verify the local existence of solution.Then constructing a new auxiliary function and using a series of inequalities,the sufficient conditions of blowup are obtained.The bounds of blowup time are estimated by using calculus theory.Through the qualitative study of the solution for these weighted nonlocal diffusion equations and systems,we find that compared with the local diffusion model,both the local existence and the blowup properties of solutions are affected by the nonlocal diffusion term and the weighted term.In addition,different from the upper and lower solution method adopted in local diffusion,this thesis mainly adopts the method of constructing new auxiliary functions to study the blowup properties of the solution,simplify the calculation.