Reaction Diffusion System With Nonlocal Boundary And Nonlocal Initial Conditions

In the chapter there of this paper, we consider the uniqueness and the existence of solutions of the following reaction diffusion system with nonlocal boundary conditions. The following two theorems were obtained Theorem 3.1.1 Let the condition(H1-H3)be satisfied, let U ( x , t ) = (u₁| ( x , t ), u₂| ^{x , t}), V ( x , t ) = (^|u₁ ( x , t ), ^|u₂( x , t))be a pair of ordered upper and lower solutions of ( P₁ ) ,and V ( x,t)≤ U(x,t)be satisfied, let ∑_V^U ={u|u = (u₁ , u₂) ∈R²,satisfying V ( x , t ) ≤ u ≤ U ( x , t ),( x , t) ∈Ω|-_T}, Then problem ( P1 )have solutions in ∑_V^U,.Theorem 3.2.1 Let the condition of theorem3.1.1 be satisfied, moreover, let the condition(H4)be satisfied, then the solution is unique in UV.∑ In the chapter four of this paper, we also consider the uniqueness and the existence of solutions of the following reaction diffusion system with nonlocal boundary conditions and nonlocal initial conditions The following two theorems were obtained Theorem4.1.1 Let the conditions(A),(B),(C) be satisfied, let U = (u₁| , u₂| ), V =(^|u₁ , ^|u₂)be a pair of ordered upper and lower solutions of ( P₂ ), Then problem ( P₂ )has a maximal solution u|-= (u₁|-, u₂|-) and a minimal solution -|u = ( -|u₁ ,-|u₂)in ∑_V^U. Moreover, (0) ( ) ( 1) ( 1) ( ) (0)( 1,2), 1,2,3k k k kui ui ui ui ui ui ui ui ui ui inDTi k= ≤ ≤ + ≤ ≤ ≤ +≤ ≤ == ? =% L % lim ( ) ( , ) ( , ),lim ( )( , ) ( , ),( 1,2)k kk ui t x ui t x kui t x ui t xi→∞ = →∞== Theorem4.2.1 Let the condition(A),(B),(C),and the condition(4.5),(4.6) be satisfied, let U = (u %1 , u% 2 ), V =(u % 1 , u% 2)be a pair of ordered upper and lower solutions of ( P2 ), Thenu = uis a unique solution of problem ( P2 ) in UV.∑...