Research Article The Existence and Behavior of Solutions for Nonlocal Boundary Problems

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1 Hindawi Publishing Corporation Boundary Value Problems Volume 9, Article ID , 17 pages doi:1.1155/9/ Research Article The Existence and Behavior of Solutions for Nonlocal Boundary Problems Yuandi Wang 1 and Shengzhou Zheng 1 Department of Mathematics, Shanghai University, Shanghai 444, China Department of Mathematics, Beijing Jiaotong University, Beijing 144, China Correspondence should be addressed to Yuandi Wang, yuandi wang@yahoo.com.cn Received 16 October 8; Revised 19 March 9; Accepted 3 March 9 Recommended by Pavel Drabek The purpose of this work is to investigate the uniqueness and existence of local solutions for the boundary value problem of a quasilinear parabolic equation. The result is obtained via the abstract theory of maximal regularity. Applications are given to some model problems in nonstationary radiative heat transfer and reaction-diffusion equation with nonlocal boundary flux conditions. Copyright q 9 Y. Wang and S. Zheng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction The existence of solutions for quasilinear parabolic equation with boundary conditions and initial conditions can be discussed by maximal regularity, and more and more works on this field show that the maximal regularity method is efficient. Here we will use some of recently results developed by H. Amann to investigate a specific boundary value problems and then apply the existence theorem to two nonlocal problems. This paper consists of three parts. In the next section we present and prove the existence and unique theorem of an abstract boundary problem. Then we give some applications of the results in Sections 3 and 4 to two reaction-diffusion model problems that arise from nonstationary radiative heat transfer in a system of moving absolutely black bodies and a reaction-diffusion equation with nonlocal boundary flux conditions.. Notations and Abstract Result We consider the following quasilinear parabolic initial boundary value problem IBVP for short : u t A t, x, u u f t, x, u, u, in Q T,

2 Boundary Value Problems B t, x, u u δg t, x, u, u, on, u x, u x, on,.1 where is a bounded strictly Lipschitz domain with its boundary 1 and 1, Q T,T, A t, x, u u a t, x, u u,. and A is a second-order strongly elliptic differential operator with the boundary operator given by B t, x, u u : δ νa u 1 δ γu..3 The coefficient matrix a a ij n n satisfies regularity conditions on Q T R, respectively. The directional derivative νa u : γa u ν, ν is the outer unit-normal vector on ; the function δ : {, 1} is defined as δ 1 j : j for j, 1; γ denotes the trace operator. We introduce precise assumptions: f : Q T R n 1 R, g : R R n 1 R,.4 where R :, are Carathéodory functions; that is, f resp., g is measurable in t, x Q T resp., in t, x, y R for each u R and continuous in u for a.e. t, x Q T resp., t, x R. More general, the function g can be a nonlocal function, for example, g t, x, u k t, x, y, u dy or g t, x, u k t, x, u dσ. Let X and Y be Banach spaces, we introduce some notations as follows: i J T :,T, J T : J T \{}. α β : min{α, β}, α β : max{α, β}. ii D D, Y : {φ C D,φ : D Y, supp φ D} for D R l, D 1 : {v D J T } {v }. iii L X, Y : {all continuous linear operators from X into Y}, andl X : L X, X. iv f t, u denotes the Nemytskii operator induced by f t, x, u t, x, u t, x. v C 1 X, Y denotes the set of all locally Lipschitz-continuous functions from X into Y.

3 Boundary Value Problems 3 vi Car,λ,λ M R R n, R, λ, andλ 1, denotes the set of all Carathéodory functions f on m M such that f m,, and there exists a nondecreasing function ψ with f m, u, ξ f m, v, ξ ψ ( ρ )( 1 ξ λ 1) u v, ( ) ( ) ( f m, u, ξ f m, u, η ψ ρ 1 ξ λ 1 ) η λ 1 ξ for u, v ρ..5 η Particularly, f is independent of ξ if λ λ 1. vii W s p denotes the Sobolev-Slobodeckii space for s R and p 1 with the norm W s p, especially, W p L p ;and W s p : W s p : W s 1/p p W s 1 1/p p 1 ( p>1 )..6 viii W s, s, \{Z 1/p} Z is the set of integral numbers, is defined as p,b W s q,b { } u Wq; s Bu, 1 1 <s, q } {u Wq; s 1 γu on, q <s<1 1 q, Wq, s s< 1 q, ( ), W s 1 q,b s<, s/ Z q,.7 where q q/ q 1, X is the dual space of X, andb is the formally adjoint operator. x W 1 p J : W 1 p J, E 1,E : Wp J, 1 d E L p J, E 1 if E 1 E and J is an interval in R. xi MR p J T : MR p J T, E 1,E denotes all maps B possessing the property of maximal L p regularity on J T with respect to E 1,E, that is, given h L p J T,E, the initial problem v Bv h, in J T, v.8 has a unique solution v W 1 p J T, E 1,E. Now we turn to discuss the local existence result. We write E 1 W s p,b, E W s p,b, E E 1/p,p W s /p p,b,.9

4 4 Boundary Value Problems then, C J T,E, W 1 p J T, E 1,E ) C (Q T, if p>n..1 Exactly, W 1 p J T, E 1,E BUC Q T as p>n, where BUC Q T denotes the Banach space of all functions being bounded and uniformly continuous in Q T. So, we will not emphasize it in the following. A weak solution u of IBVP.1 is defined as a W 1 p J T, E 1,E function u, s 1, 1 1/p, satisfying T v, u v, a t, u u dt v,u T { v, f t, u γv,g t, u 1 } dt v D1,.11 where, and, j denote the obvious duality pairings on and j, respectively. Set f t, u : f t f 1 t, u,g t, u : g t g 1 t, u with f 1 t, g 1 t,..1 After these preparations we introduce the following hypotheses: H1 p>n ands 1, 1 1/p. H a t, x, u C,α,1 J T R, R n n with α 1 >s, and there exists a δ, 1 such that δ ξ ξ a t, x, u ξ ξ /δ, t, x, u J T R..13 f,g L p J, E Wp, s f 1 Car,λ,λ 1 Q T R R n with λ 1,, and λ 1 < 1 p s 1 / p 1 s. H3 g 1 t, C 1 W 1 p J T,L r J T, Wp s for some r>p. Theorem.1. Let assumptions (H1) (H3) be satisfied. Then for each u E the quasilinear problem.1 possesses a unique weak solution u t, x W 1 p J T, E 1,E for some T >. Proof. Recall that E 1 C s n/p( ), E C s n /p( )..14 The Nemytskii operator a,u is defined as a,u t, x : a t, x, u t, x. The fact a,u t, x C,α s n /p ( Q T ).15

5 Boundary Value Problems 5 shows the maximal regularity of the operator A. By 1, Theorem.1, if, for t T, f 1 C 1 W 1 p J t,l r J t,e for some r > p, then the existence and the uniqueness of a local solution will be proved. The remain work is to check the Lipschitz-continuity. Set q : np n s p, θ : 1 p.16 1 s. Then L q E.So,foru, v W 1 p J t with u, v ρ we have f1 t, x, u, u f 1 t, x, u, v Lr J t,e f1 t, x, u, u f 1 t, x, u, v Lr J t,l q c ( ρ ) u λ 1 1 u v Lr J t,l q c ( ρ ) u λ 1 1 L λ1 1 pq/ p q u v L p Lr J t..17 From λ 1 1 q <p q, we infer that f 1 t, x, u, u f 1 t, x, u, v Lr J t,e c ( ρ ) u λ 1 1 u v Wp 1 Wp 1 c ( ρ ) u λ θ C I t,e u v 1 θ Lr J t C I t,e u λ 1 1 θ E 1 u v θ E 1 Lr J t.18 c ( ρ ) u λ θ C I t,e u v 1 θ C I t,e u θ λ 1 1 L r J t,e 1 u v θ L p J t,e 1, where r : λ 1 1 prθ/ p θr. Notethatλ 1 θ<1, we can choose r>psuch that f1 t, x, u, u f 1 t, x, u, v ( ) Lr J t,e c u W 1 p J t u v W 1 p J t..19 On the other hand, the hypotheses guarantee that f t, x, u, v f t, x, v, v Lr J t,e f t, x, u, v f t, x, v, v Lr J t,l q c ( ρ ) v λ 1 u v Lr J t,l q c ( ρ ) u v C Jt,E v λ 1 Lr J. t,l q.

6 6 Boundary Value Problems Due to q<pand λ 1,,Hölder inequality follows that f t, x, u, v f t, x, v, v Lr J t,e c( ρ ) ( t ) 1/r. u v C Jt,E v λ 1 r L p dτ.1 The hypothesis of λ means that one can find an r>psuch that f t, x, u, v f t, x, v, v ( ) Lr J t,e c ρ, v W 1 p J t u v W 1 p J t.. Obviously, if λ 1, the above inequality is followed from. immediately. Hence it follows from.19 and. that ( ) f t, x, u, u f t, x, v, v Lr J t,e c u, v W 1 p J t u v W 1 p J t..3 This ends the proof. We apply the above theorem to the following two examples in next sections. For this, in the remainder we suppose that hypotheses H1 - H hold and that 1, ṗ : n 1 p n 1 s p A Radiative Heat Transfer Problem We see a nonlinear initial-boundary value problem, which, in particular, describes a nonstationary radiative heat transfer in a system of absolutely black bodies e.g., refer to. A problem is u t a t, x, u u f t, x in Q T, νa u u h ( u ( t, y )) ϕ ( t, x, y ) dσ ( y ) h u t, x g t, x on,t, 3.1 u,x u x, on Local Solvability We assume that Hr Hr1 ϕ L r J T,Lṗ ; Hr h is locally Lipschitz continuous and h. Theorem 3.1. Let assumptions (H1)-(H) and (Hr) be satisfied. Then problem 3.1, for all u E, has a unique u W 1 p J T for some T >.

7 Boundary Value Problems 7 Proof. Note that the embedding.14 holds: Lṗ W s p, (. Lṗ W 1/p s p W s 1/p p ) 3. Hence Theorem.1 implies the result immediately. In fact, Amosov proved in 5 the uniqueness of the solution for a simple case, that is, problem in which the matrix a is independent of u see, Theorem 1.4. In this paper, we also can get the positivity of the solution and the estimates of the solution in W 1 and L in this part. We have tried to achieve the global existence, but it is still an open problem. In the rest of this section, we always assume that H1 - H and Hr hold. 3.. Positivity Assume that H h u is nondecreasing with h, and ϕ ( t, x, y ) ϕ ( t, y, x ), ϕ t, x : ϕ ( t, x, y ) dσ ( y ) < Theorem 3.. Let assumption (H ) be satisfied. If f,g,u is nonnegative, then the solution u of problem 3.1 is also nonnegative. Proof. Put u : u. Multiplying the equation with u and integrating over, we have u dx f 1 ( ) u dx t 1 ( ) u dx t 1 ( ) u dx t [ h u t, x a u u u dx a u u u dx a u u u dx u u dσ ν a h ( u ( t, y )) ϕ ( t, x, y ) dσ ( y )] u dσ g u dσ. By using the assumption of H, we can get following equality: 3.4 h ( u ( t, y )) ϕ ( t, x, y )( u ( t, y ) u t, x ) dσ ( y ) dσ [ ( ( ))] ( )( ( )) ( ) h u t, x h u t, y ϕ t, y, x u t, x u t, y dσ y dσ.

8 8 Boundary Value Problems So, [ h u t, x h ( u ( t, y )) ϕ ( t, x, y ) dσ ( y )] u t, x dσ h u t, x u t, x ( 1 ϕ t, x ) dσ h ( u ( t, y )) ϕ ( t, x, y )( ( ) u t, y u t, x ) dσ ( y ) dσ h u t, x u t, x ( 1 ϕ t, x ) dσ 1 [ ( ( ))]( ( )) ( ) ( ) h u t, x h u t, y u t, x u t, y ϕ t, y, x dσ y dσ. 3.6 At the last inequality, the monotonity of h on u and the restriction ϕ <1 are used. Therefore, d dt u L f u dx g u dσ. 3.7 If u, then u t, x. The assertion follows W 1 -norm We denote by J max the maximal interval of the solution of problem 3.1. Lemma 3.3. There exists a constant C C T; f,g,u such that the solution u of problem 3.1 satisfies u t, L u L Q t C for t J max. 3.8 Proof. Multiplying by u and integrating over, we have u t a u u u dx f udx. 3.9 That is, 1 d u dx a u u udx u νa udσ f udx. dt 3.1

9 Boundary Value Problems 9 As similar as the inequality 3.6, we have [ u νa udσ u h ( u ( t, y ) ϕ ( t, x, y ) dσ ( y ) h t, x ) g ]dσ ( ) 1 ϕ t, x h u t, x udσ g udσ 1 { ( ( ))}[ ( )] ( ) h u t, x h u t, y u t, x u t, y ϕ t, x, y dσ dσ g udσ Hence, 1 d u dx δ u dx f udx dt { ɛ u dx g udσ u dσ } 1 { f ɛ dx g }. dσ 3.1 By using the embedding W 1,B L and letting ɛ small enough, it is easy to get that u L u L Q t C( T; f,g,u ), for t Jmax L -norm Theorem 3.4. If f L Q T and g L,T, then the solution u t, x of problem 3.1 is bounded with its L -norm for all t J max. Proof. From the hypothesis H1 and embedding.1, one has that u C Q T and u n W 1, n 1,,... By multiplying with uk 1 k Z and k and integrating over,we have u t a u u u k 1 dx f u k 1 dx That is, k d dt u k dx a u u u k 1 dx u k 1 νa udσ f u k 1 dx. 3.15

10 1 Boundary Value Problems But, ( ) a u u u k 1 dx k 1 u k a u u udx u k 1 νa udσ ( k 1 ) k a u u k 1 u k 1 dx, [ u k 1 h ( u ( t, y )) ϕ ( t, x, y ) dσ ( y ) h t, x g ]dσ ( ) 1 ϕ t, x u k 1 h u t, x udσ g u k 1 dσ 1 { ( ( ))} [ h u t, x h u t, y u k 1 t, x u k 1 ( t, y )] ϕ ( t, x, y ) dσ, g u k 1 dσ Therefore, 1 d ) u k dx δ k ( k 1 u k u dx dt 1 k ɛ d k 1 u k dx δ dt 4 k 1 fu k 1 dx ( g u k 1 dσ 1 k){ u k dx ɛ 1 k k { u k 1 dx } u k dσ f k dx g k }, dσ 3.17 where Young s inequality, αβ ɛ/r α r ɛ r /r /r β r α, β, <ε<1, has been used at the last inequality. We apply the embedding W 1,B L again with v u k 1 and choose ɛ small enough, then we attain the following inequality: d { } { u k dx ɛ C k ɛ dt u k dx ɛ 1 k f k dx g k }. dσ 3.18 By Gronwall s inequality, the inequality 3.18 becomes t { u k dx e ɛ C k ε t u k dx ɛ1 k e ɛ C k ɛ t τ f k dx g k }dτ. dσ 3.19

11 Boundary Value Problems 11 Set B : 1 u L f L Q T g L,T, then we deduce that u L k B e ε C t [e ɛt ε εu k B k dx ɛ t { f k B k dx g k B k } dσ dτ ] 1/ k 3. ε 1 B e ε C t t 1/k. Let k, the inequality 3. implies u t, L e ɛ C t u L C (T; f L Q T, g L,T ) for t J max. 3.1 The claim follows. One immediate consequence of the above theorem is. Corollary 3.5. The L -norm of the solution u, that is, u t, L, of problem 3.1 is nonincreasing if f g. 4. A Nonlocal Boundary Value Problem We now consider the problem.1 with the following boundary value condition: B t, x, u g t, x, u, u Φ u t, x g 1 t, x, u g. 4.1 The function Φ in 4.1 can be in nonlocal form. IBVP.1 with a nonlocal term stands, for example, for a model problem arising from quasistatic thermoelasticity. Results on linear problems can be found in 3 5. As far as we know, this kind of nonlocal boundary condition appeared first in 195 in a paper 6 by W. Feller who discussed the existence of semigroups. There are other problems leading to this boundary condition, for example, control theory see 7 1 etc.. Some other fields such as environmental science 13 and chemical diffusion 14 also give rise to such kinds of problems. We do not give further comments here. Carl and Heikkilä 15 proved the existence of local solutions of the semilinear problem by using upper and lower solutions and pseudomonotone operators. But their results based on the monotonicity hypotheses of f, g, andφ with respect to u. In this section, we assume that H1 and H always hold and assume that Hn1 Φ andφ C 1 W 1 p J T,L r J T,Lṗ for some r>p; Hn g 1 t, x,, g 1 satisfies the Carathéodory condition on t, x and g 1 t, x, C 1 R. By the embedding theorem and Theorem.1, we get immediately. Theorem 4.1. Suppose hypotheses of (Hn) satisfy. Then problem.1, for all u E,withg defined in 4.1 has a unique u W 1 p J T for some T >.

12 1 Boundary Value Problems For the simplicity in expression, we turn to consider a problem with nonlocal boundary value u t A t, x, u u f t, x, u, u, in Q T, B t, x, u u κ u t, x g 1 t, x, u g, u x, u x, on, on, 4. where κ u t, x : k ( t, x, y, u ( t, y ), u ( t, y )) dy, 4.3 and Hk The function k satisfies the Carathéodory condition on t, x, y Q T :,T,k u and f k Car,λ, λ Q T R R n with λ < 1 p s 1 p 1 s, λ <p Theorem 4.. Let assumption (Hk) be satisfied. Then Problem 4., for any u E, has a unique solution u W 1 p J T for some T >. Proof. First, we see that u λ 1 u v dy Lr J t u λ 1 L λ 1 p u v L p c u λ 1 u v W 1 p Lr J t. W 1 p Lr J t 4.5 Choose θ, 1 such that λ 1 θ<1, then λ 1 θ/ 1 θ < 1. Consequently, there exists r>psuch that u λ 1 u v dy Lr J t c u λ 1 1 θ C J t u v 1 θ,e C J t,e ( ) c u W 1 p J t u v W 1 p J t. u λ 1 pθ/ p θr E 1 Lr u v θ Lr E J t 1 J t 4.6

13 Boundary Value Problems 13 Similarly, from λ p 1 we have u λ 1 u v dy Lr J t ( t c u v C Jt,E u r dτ Wp 1 ( ) c u W 1 p J t u v W 1 p J t. ) 1/r 4.7 Combining two inequalities 4.6 and 4.7, weobtainthat κ u κ v Lr J t, Wp s k t, x, y, u, u k t, x, y, v, v dy c Lr J t,lṗ [( ) ( ) ] ψ ρ 1 u λ 1 v λ 1 u v 1 1 v λ u v ) ( c u W 1 p J t dy Lr J t u v W 1 p J t. 4.8 The claim follows immediately from Theorem 4.1. A special case of problem 4. is u t a u u f t, x, u, in Q T, νa u u k ( t, x, y, u ) dy g t, x, u, u,x u x, on. on,t, 4.9 That is, f and k in 4.9 are independent of gradient u W 1 -norm In order to discuss the global existence of solution, in the rest of this section we assume the following. Hkl Suppose there exists a continuous function φ : R R such that f 1 t, x, u, k ( t, x, y, u ), g 1 t, x, u φ t u. 4.1

14 14 Boundary Value Problems Lemma 4.3. There exists a constant C C T; f,g,u such that the solution of problem 4.9 satisfies u L u L Q t C, for t J max Proof. We multiply the first equation in 4.9 with u and then integrate over, and we find that 1 d dt u L δ u L [ f t, x, u udx u k ( t, x, y, u ) ] dy g t, x, u dσ x { ( )} ) φ t u L u L 1 u L1 u L 1 ε 1 ( u L u L 1 ( f ε L ) g L Since W θ L for θ 1/, 1, by interpolation inequality and Young s inequality we have that u L C u W θ C u θ W 1 u 1 θ L 4.13 ε u W 1 C ε u L. Apply Young s inequality again and then choose ε j small enough j 1, ; itisnotdifficult to get 1 d dt u L C ε φ t u L ( δ ε ( φ t 1 )) u L 1 ( f ε L ) g L, where δ ε φ t 1 > fort, T. Therefore, by multiplying with e C ε t φ τ dτ and integrating over, t, the inequality 4.14 follows the claim. 4.. L -norm Lemma 4.4. Let assumptions of Lemma 4.3 be satisfied. If f,g L Q T,T, then the solution u of problem 4.9 satisfies u L C( T; f,g,u ), for t Jmax. 4.15

15 Boundary Value Problems 15 Proof. We multiply the first equation in 4.9 with u k 1 and integrate over, then we reach that J t : 1 d k dt u k L k δ ( k 1 ) u k 1 4 k 1 f t, x, u u k 1 dx L [ u k 1 k ( t, x, y, u ) ] dy g t, x, u dσ x ( φ t { u k L k 1 u k L k f u k 1 dx As the same as the inequality 4.13, we have g u k 1 dσ. )} u L1 k u L k 4.16 u k L k ε 1 u k 1 C ε 1 u k L L k, ( u k 1 u L k u L 1 ε k 1 ) 1 k 1 C ε 1 u k L L k u L1 ε 1 k 1 u k 1 1 k u L1 L 4.17 C ε 1 u k L k. Hence, J t { φ t 1 C ε 1 C ε } u k L k ε 1 ε u k 1 C ε k{ f k L ε1 k 1 k L k g k L k }. u k 1 1 k L 4.18 We might as well assume that u L >, so, u k 1 [ ] 1 k k 4 1 k k u k u dx u 1 L L as k The boundedness of solution u L C for t J max is used in above deduction. Let ε j j 1, small enough, then we have k d dt u k L k C ε φ t u k L k C ε k{ f k L k } g k. 4. L k

16 16 Boundary Value Problems Multiplying with k e C ε k t φ τ dτ, then integrating over, t, weobtainthat ( u Lk e C ε t φ τ dτ) k t u k L k C ε e C ε k s φ τ dτ{ f k L k } g k ds. L k 4.1 By a similar limitation process as in 3.1, weget u L C( ) T; f,g,u for t J max. 4. This closes the end of proof Decay Behavior In order to investigate the decay behavior of solution for problem 4.9, we assume that Hkd there are two continuous function ϕ t > andɛ t (t ) such that f 1 t, x, u u ϕ t u, g 1 t, x, u u, ( ) k t, x, y, u ɛ t u 4.3 for all t, x, y R. Theorem 4.5. Let the assumption (Hkd) be satisfied and, u be the solution of problem 4.9 with f,g. Then u L decay to zero as t for some small functions ɛ t. Proof. We use u to multiply the first equation in the system 4.9 and then integrate over. Thus, we get that 1 d dt u L δ u L [ f t, x, u udx u k ( t, x, y, u ) ] dy g t, x, u dσ 4.4 ϕ t u L ɛ t u L 1 u L { } Cɛ t ϕ t u L u Cɛ t W 1. In the above process the inequality 4.13 is used. If we choose ɛ t as ɛ t { } 1 C min δ, min ϕ t, t, T, 4.5 t, T

17 Boundary Value Problems 17 then u L u L e t ϕ τ dτ, t, T. 4.6 This ends the proof. Moreover, one can verify that u Lp also decay to zero as t if p. Acknowledgments The first author wishes to thank Professor Herbert Amann for many useful discussions concerning the problem of this paper. The author also want to thank the referees suggestions. This work is supported partly by the National NSF of China Grant nos and and by Shanghai Leading Academic Discipline Project no. J511. References 1 H. Amann, Quasilinear parabolic problems via maximal regularity, Advances in Differential Equations, vol. 1, no. 1, pp , 5. A. A. Amosov, Global solvability of a nonlinear nonstationary problem with a nonlocal boundary condition of radiative heat transfer type, Differential Equations, vol. 41, no. 1, pp , 5. 3 W. A. Day, A decreasing property of solutions of parabolic equations with applications to thermoelasticity, Quarterly of Applied Mathematics, vol. 4, no. 4, pp , A. Friedman, Monotonic decay of solutions of parabolic equations with nonlocal boundary conditions, Quarterly of Applied Mathematics, vol. 44, no. 3, pp , L. A. Muraveĭ and A. V. Filinovskiĭ, On a problem with nonlocal boundary condition for a parabolic equation, Mathematics of the USSR-Sbornik, vol. 74, no. 1, pp , W. Feller, The parabolic differential equations and the associated semi-groups of transformations, Annals of Mathematics, vol. 55, no. 3, pp , J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications. Vol. I,Springer, Berlin, Germany, J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications. Vol. II, Springer, Berlin, Germany, I. Lasiecka, Unified theory for abstract parabolic boundary problems a semigroup approach, Applied Mathematics and Optimization, vol. 6, no. 1, pp , H. Amann, Feedback stabilization of linear and semilinear parabolic systems, in Semigroup Theory and Applications (Trieste, 1987), P. Clement, S. Invernizzi, E. Mitidieri, and I. I. Vrabie, Eds., vol. 116 of Lecture Notes in Pure and Applied Mathematics, pp. 1 57, Dekker, New York, NY, USA, R. P. Agarwal, M. Bohner, and V. B. Shakhmurov, Linear and nonlinear nonlocal boundary value problems for differential-operator equations, Applicable Analysis, vol. 85, no. 6-7, pp , 6. 1 A. Ashyralyev, Nonlocal boundary-value problems for abstract parabolic equations: well-posedness in Bochner spaces, Journal of Evolution Equations, vol. 6, no. 1, pp. 1 8, V. Capasso and K. Kunisch, A reaction-diffusion system modelling man-environment epidemics, Annals of Differential Equations, vol. 1, no. 1, pp. 1 1, K. Taira, Diffusion Processes and Partial Differential Equations, Academic Press, Boston, Mass, USA, S. Carl and S. Heikkilä, Discontinuous reaction-diffusion equations under discontinuous and nonlocal flux conditions, Mathematical and Computer Modelling, vol. 3, no , pp ,.