Note Of Solutions For A Class Of Quasilinear Reaction-Diffusion Systems And Corresponding Elliptic Systems

By using method of supersolution and subsolution,this paper deals with the solution either exists globally or blows up in finite time to a class of quasilinear reaction-diffusion systems with nonlocal sources.The conditions of existence of global solution and blow up in finit time are given respectively.Meanwhile by using the Schauder fixed point theory and integration method, we as well as study the existence,uniqueness in ball B(R) and the nonexistence in R~N of positive radial solutions of the corresponding elliptic systems with no non-local sources when the reaction-diffusion systems reach the balance condition.The main content of this paper is divided into two chapters.In chapter two ,wo discuss the solution either exists globally or blows up in finite time to a class of quasilinear reaction-diffusion systems with nonlocal source.where is a bounded domain in R~N(N ≥ l)with sufficiently smooth boundary ., .The initial datas are continuous and bounded.where I is n identity matrix.Our main results are as follows.(1) (Local Existence)Give ,there is some 0such that there ex-ists a nonnegative weak solution for each T < T*.Furthermore T* = ∞ or (2) (Uniqueness)The solution()oi (*) is nuiquely determined by the initial data (3)(Global existence)(i)If A is a nonsingular M-matrix,i.e all the principal minors of A have postive determinants ,then the solution of (*)exists globally.(ii)If A is a singular M-matrix,i.e each principal sub-matrix of A other than A itself is a nonsingular M-matrix,and is sufficiently small.then the solution of (*)exists globally.(iii)If A is not an M-matrix ,i.e not all the principal minors of A have positive determinants, and the initial data is sufficiently small,then the solution of (*)exists globally.(4) (Blowing up in finite time)(i)If A is not an M-matrix ,i.e not all the principal minors of A have postive determinants, and the initial data is sufficiently large,then the solution of (*)blows up in finite time.(ii)If A is a singular M-matrix,i.e each principal sub-matrix of A other than A itself is a nonsingular M-matrix,and is sufficiently large,then the solution of (*)blows up in finite time.By the statements as above,we can find out that whether or not A is a so-called M-matrix plays an important role in the solution studying of the reaction-diffusion systems.Otherwise, it has close relation to the solution existence of the elliptic systems.In chapter three ,we discuss the existence,uniqueness in ball B(R)and the nonexistence in the corresponding elliptic systems with no non-local sources.We obstain the following results.(l)If A (as above)is a nonsingular M-matrix,then for the initial 0 ,system has postive radially solution.(2)Under the condition of (l),the solution of (**)is unique. (3)Give if min ,then the system(**)in R~N has no postive radially solution, where...