Integro-differential systems with variable exponents of nonlinearity

Abstract Some nonlinear integro-differential equations of fourth order with variable exponents of the nonlinearity are considered. The initial-boundary value problem for these equations is investigated and the existence theorem for the problem is proved.


Notation and statement of theorem
Let jj jj B Á jj I Bjj be a norm of some Banach space B, B N WD B : : : B (N times) be the Cartesian product of the B, B be a dual space for B, and h ; i B be a scalar product between B and B. We use the notation X « Y if the Banach space X is continuously embedded into Y ; the notation X _ « Y means the continuous and dense embedding; the notation X .7/ and jjzI B N jj WD jjz 1 I Bjj C : : : C jjz N I Bjj.

Properties of generalized Lebesgue and Sobolev spaces
The following Propositions are needed for the sequel.

Auxiliary functional spaces
Let L.X; Y / be a space of bounded linear operators from X into Y (see [33, p. 32 . / is reflexive. Let fw j g j 2N be a set of all eigenfunctions of the problem w j D j w j in ; w j j @ D 0; j 2 N: .28/ Here f j g j 2N R C is the set of the corresponding eigenvalues. Suppose that fw j g j 2N is an orthonormal set in L 2 . /. It is easy to verify that solutions to problem (28) satisfy the equalities wj @ D wj @ D : : : D r 1 wj @ D 0: .29/ The following propositions are needed for the sequel.  where r is determined from condition (Z). We consider the space V N (see (14) ) with respect to the norm Since r satisfies (Z) and (14) holds, it is easy to verify that The following Lemma is needed for the sequel. The proof is omitted (see for comparison Lemma 8.1 [35, p. 307]). We consider the space U.Q 0;T / (see (15) ) with respect to the norm It is easy to verify that the space U.Q 0;T / is reflexive. Taking into account the embedding of type (25) and inequality (30), we obtain The proof is omitted (see for comparison [36, p. 5] and [13,27]).

Projection operator
Let H be the Hilbert space and V be the reflexive separable Banach space such that Since v; w 1 ; : : : .42/ iff the following equality holds Proof. Clearly, (43) implies (42). We shall prove that (42) implies (43). Take v 2 V. There exist numbers m 1 ; : : : w . Multiplying both sides of -th equality of (42) by m and summing the obtained equalities, we get Taking into account (37), the inclusions z m ; w 1 ; : : : ; w m 2 V, and the orthonormality condition for fw j g j 2N H, Therefore, (42) yields (43).

Differentiability of the nonlinear expressions
Take a function 2 M. / and by definition, put The function v WD maxf u; 0g has a similar property.
The following Propositions are needed for the sequel.
.53/ holds if one of the following alternatives hold:  (55) yields (53). We shall omit the proof of (ii) because it is analogous to the previous one.
Note that the case .x/ Á 2 .0; 1 is considered in [45].      2) there exists a weak solution to problem (74) which is defined on right maximal interval of existence OE0; b/, where b Ä T . We shall prove that Case 2 is impossible. Assume the converse. Then for every 2 .0; b/ this local weak solution ' belongs to W 1;p .0; I R`/. Define .80/ where˛andˇare determined from (78). Since L satisfies the L p -Carathéodory condition and R is determined from (80), there exists a function h R 2 L p .0; T / such that for a.e. t 2 .0; T / and for every 2 D R WD fy 2 R`j jyj Ä Rg inequality (75) holds. Taking into account (see (78) ) the following inequalities from (74) we get Therefore, from (77) we get where R is determined from (80). Thus j'.
Step 3. Taking into account the results of Step 1 and Step 2, we obtain that the function I satisfies the Carathéodory condition. Since g 2 L 1 .Q 0;T /, the L 1 -Carathéodory condition holds. .86/ where C 14 > 0 is independent of u and .
Step 3. Taking into account the results of Step 1 and Step 2, we obtain that the function J satisfies the Carathéodory condition. Since 2 L 1 .Q 0;T /, the L 1 -Carathéodory condition holds.
Clearly, the operator ƒ.t / W Z N ! OEZ N (see (16) .89/ where S 1= 0 and S are defined by (8), C 19 > 0 is independent of u, v and t.
Proof. Similar to [54, p. 159], we use the generalized Hölder inequality, Proposition 3.3 with q D , and notation (7). We get the estimate Since 0 Ä 2, we obtain that (89) holds and the operator ‰ is bounded. We omit the proof that ‰ is semicontinuous (it is similar to the proof of Lemma 3.25).
Let us consider the Banach space V such that V « Z N . Let us define the family of operators By (89), we obtain .90/ where C 21 > 0 is independent of u, v and t . Then ‰ V W V ! V is bounded. We will replace this space V by V N and W r . For the sake of convenience we have replaced ‰ V N and ‰ W r by ‰ and we have replaced h ; i V N and h ; i Wr by h ; i. The same notation we need for ƒ.t /, A.t /, and K.t /, t 2 .0; T /. According to the above remarks, we have that the operator K.t / (see (19) ) is bounded and semicontinuous from V N into OEV N and is bounded from W r into W r . Proof. Clearly, .94/ Using the generalized Young inequality, we get

Proof of main Theorem
The solution will be constructed via Faedo-Galerkin's method.
Step 1. Let fw j g j 2N be a set of all eigenfunctions of the problem (28)  r is determined from condition (Z), W r and W r are defined by (31), and V is defined by (14). Taking into account .101/ where C 29 ; C 31 > 0 are independent of t; ' m . Then Carathéodory-LaSalle's Theorem 3.24 implies that there exists a solution ' m 2 H 1 .0; T I R mN / to problem (97), (98). If we combine the condition @ 2 C 2r with Proposition 3.5 and embedding (26), we get fw j g j 2N W r OEH 2r . / N . Thus, .u m t ; u m / dxdt D where C 32 ; C 33 > 0 are independent of m and . In addition, Young's inequality, the condition @ 2 C 2 , and estimate (30) yield thať where~1 > 0, the constant C 34 > 0 is independent of m and~1. By (23), we geť According to the above remarks, from (103) we have the following inequality ju m x i j p.x/ 1 jv x i j C ju m j .x/ 1 jvj C ju m j q.x/ 1 jvj C ju m j jvj C jEu m j jvj .125/ where ' 2 C 1 0 ..0; T //, w 2 M N k , k 2 N, k Ä m j , j 2 N. Letting j ! C1 and using Lemma 3.8, we get the equality u t C Au D F. Whence, u t D F Au 2 OEU.Q 0;T / , u 2 W .Q 0;T /, and (22) holds. Moreover, we obtain the inclusion u t 2 L s 0 s 0 1 .0; T I OEV N / because (34) is true. Hence, u 2 C.OE0; T I OEV N /. By (108), we have that u 2 L 1 .0; T I H N /. Thus, Lemma 3.7 yields that u 2 C.OE0; T I H N / and so u is a weak solution to initial-boundary value problem (1), (2).