The study of a mixed problem for one class of third order differential equations

This work is dedicated to the study of existence and uniqueness of almost everywhere solution of a multidimensional mixed problem for one class of third order differential equations with nonlinear operator on the right-hand side. The conception of almost everywhere solution for the mixed problem under consideration is introduced. After applying the Fourier method, the solution of the original problem is reduced to the solution of some countable system of nonlinear integro-differential equations in unknown Fourier coefficients of the sought solution. Besides, existence and uniqueness theorems of almost everywhere solution of the mixed problem under consideration are also proved in this work.


Introduction
This work is dedicated to the study of existence and uniqueness of almost everywhere solution for the following multidimensional mixed problem: where 0 < T < +∞; x = (x 1 , . . . , x n ), is a bounded n-dimensional domain with an enough smooth boundary S; = [0, T] × S; functions a ij (x) (i, j = 1, n) and a(x) are measurable and bounded in and satisfy in the following conditions: where ξ i (i = 1, . . . , n) are arbitrary real numbers; ϕ, ψ are the given functions; is some, generally speaking, nonlinear operator, and u(t, x) is a sought function.
It must be noted that many problems in elasticity theory, in particular the problems of longitudinal vibration of the viscoelastic non-homogeneous bar, some wave problems for elastic-viscidal liquid, etc. lead to the problems of type (1)- (3).
In the beginning, we will note some papers related to problem (1)- (3). Note that the results of this work is complete progression of the results of [1]. In particular in this work we prove new theorems about existence and uniqueness of the almost everywhere solution of problem (1)- (3).
In [10] a mixed problem for the equation below was considered Under certain special conditions with respect to the nonlinear function f the existence of the solution of the problem under consideration for all t > 0 has been proved.
In [9], a mixed problem for the equation with a nonlinearity of the type |u| p-1 . u was considered. The conditions for the existence of a global weak solution of this problem are indicated. Further, in [8] and [2] a special case of problem (1)-(3) is considered, when the operator , appearing on the right-hand side of equation (1), is an operator of the type of a function generated by the function f (t, x, u, u t , ∇u, ∇u t , ∇ 2 u). In [8] by combining the generalized contracted mappings principle and Schauder's fixed point principle for any dimensions n, the existence in small theorem (i.e., for sufficiently small values of T) and the uniqueness in large theorem (i.e., for any finite value of T) of almost everywhere solution of problem (1)-(3) was proved, and using the method of a priori estimates for all dimensions n, the existence in large theorem of almost everywhere solution of problem (1)-(3) was proved. But in [2] the authors investigated the existence in small of classical solution of problem (1)-(3), and using the contracted mappings principle for any dimensions n, the existence theorem in small of classical solution of the considered mixed problem was proved.
In the work [7] the existence and uniqueness of the strong global solution of the one special case of problem (1)-(3), when L = , n ≤ 3, and = u + f (u), were proved.
Finally, we mention the work [5] in which theorems about existence and uniqueness of the generalized, almost everywhere, and classical solution for one special one-dimensional case of problem (1)-(3), when n = 1, = (0, 1), Lu = α · u xx , were proved.

Auxiliaries
In this section, we introduce a number of concepts, notations, and facts to be used later.
We denote byḊ( ) the class of all continuously differentiable functions on which vanished near the boundary of . We denote the closure ofḊ( ) with respect to the norm of W 1 2 ( ) byD( ). HenceD( ) ⊂ W 1 2 ( ).
Denote byḊ 1 (Q T ) (Q T ≡ [0, T] × ) the class of all continuously differentiable functions on the cylinder Q T equal to zero in the δ-neighborhood of the lateral surface on the cylinder Q T , having the form Q T,δ = [0, T] × δ , where δ is a δ-neighborhood of the boundary of . We denote the closure ofḊ 1 (Q T ) with respect to the norm of W 1 2 (Q T ) byD 1 (Q T ). 2. For investigation of problem (1)-(3), we recall one property of the operator L, generated by the differential expression (4) and boundary condition (3): there are denumerable number of negative eigenvalues with the corresponding generalized eigenfunctions υ s (x) which are complete and orthonormal in L 2 ( ). We call the function υ s (x) ∈D( ) a generalized eigenfunction of the operator L if it is not identically zero and for any function (x) ∈D( ).
As the system {υ s (x)} ∞ s=1 is complete orthonormal in L 2 ( ), then it is evident that every almost everywhere solution of problem (1)-(3) has the following form: Then, after applying the Fourier method, finding the unknown Fourier coefficients u s (t) (s = 1, 2, . . .), the almost everywhere solution u(t, x) of problem (1)-(3) is reduced to the solution of the following countable system of nonlinear integro-differential equations: where Proceeding from the definition of almost everywhere solution of problem (1)-(3), it is easy to prove the following.
where s = 1, 2, . . . and τ ∈ [0, T], then with the help of integration by parts with respect to t twice in the first term and once in the second term of (6) and taking the initial conditions (2) into consideration, we easily get where Differentiating (7) three times with respect to τ , we have the next problem which is obviously equivalent to system (5). The lemma is proved.
We define the norm in this set as u = N T (u 5. We agree to assume that all the quantities throughout this work are real, all the functions are real-valued, and all the integrals are understood in the sense of Lebesgue.

On the existence of almost everywhere solution
In this section, using Zabreyko and Krasnoselskiy's fixed point principle, the following existence theorem for the almost everywhere solution of problem (1)-(3) is proved for n. where 3.
We consider the operators Q 1 (u) = W + P 1 (u) and Q 2 (u) = P 2 (u) in the closed ball K 2 ( u B 3,2 2,2,T ≤ R), where the function W (t, x) is defined by (8) and Using condition 4 of this theorem and the relation Lυ s (x) = -λ 2 s υ s (x), we transform (by integrating by parts) system (5) to the following form: Then we have, for every u ∈ K 2 , It is easy to obtain that, for any u, v ∈ K 2 , where (u(t, x)) = 1 (u(t, x)) + 2 (u(t, x)) and the number C 0 is defined by (10). From inequality (15), which holds not only for u, v ∈ K 2 , but also for all u, v ∈ K 1 , follows that the operator Q 1 acts continuously from the closed ball K 1 into B 3,2 2,2,T . Since the closed ball K 2 is imbedded into the closed ball K 1 compactly by virtue of [4, Theorem 1.1, p. 51], then the operator Q 1 acts compactly from K 2 into B 3,2 2,2,T . Further, since q 0 < 1, then from inequality (16) it is seen that the operator satisfies a Lipschitz condition on K 2 with the coefficient which is less than of unity. From inequality (14) it is seen that the operator Q = Q 1 + Q 2 acts from the closed ball K 2 into itself. Hence, by virtue of the principle of Zabreyko and Krasnoselskiy about fixed point [11, p. 1234], the operator Q has at least one fixed point u(x, t) in the closed ball K 2 , which is easy to verify (in absolutely the same way as in the proof of Theorem of [1]), which is an almost everywhere solution of problem (1)-(3). The theorem is proved.

On the existence and uniqueness of almost everywhere solution
In this section, using the successive approximations method, the following existence and uniqueness theorem for the almost everywhere solution of problem (1)-(3) is proved for n.
Then problem (1)-(3) has a unique almost everywhere solution u(t, x) in B 3,2 2,2,T , which can be found by the method of successive approximation. Moreover, the distance between the solution u(t, x) and the kth approximation u k (t, x) is estimated by the following formula: Proof We consider the following operator Q: where the function W (t, x) is defined by (8) and Then it is obvious that the operator Q acts in the space B 3,2 2,2,T and satisfies the condition in the ball K 0 , where C 0 is defined by (10).
Next, using condition 4 of this theorem and taking (24) into account, we have, for every k (k = 1, 2, . . .) and t ∈ [0, T], Consequently, From this it follows that the sequence {u k (t, x)} ∞ k=1 forms the Cauchy sequence in B 3,2 2,2,T . Due to the completeness of the space B 3,2 2,2,T , we have Since the operator Q is continuous in the ball K 0 (see inequality (22)), from the relations u k (t, x) = Q(u k-1 (t, x)) it follows that It is easy to verify (in absolutely the same way as in the proof of Theorem of [1]) that the function u(t, x) is the almost everywhere solution of problem (1) Consequently, u(t, x) ∈ B 2,1 2,2,T . As u(t, x) ∈ G ∩ B 2,1 2,2,T , then due to condition 2 of this theorem (u(t, x)) ∈ W 0,1 t,x,2 (Q T ), and due to condition 3 of this theorem for almost all t ∈ [0, T] (u(t, x)) ∈D( ). Using this, from system (13) equivalent to system (5) Consequently, u(t, x) ∈ B 3,2 2,2,T . Then, using inequality (17), similar to (23)  Thus, all the possible almost everywhere solutions of problem (1)-(3) belong to the sphere K 0 , and there is a fixed point in B 3,2 2,2,T of the operator Q defined by (21). Let u(t, x) and v(t, x) be two arbitrary almost everywhere solutions of problem (1)-(3). Then, using the fact that u(t, x), v(t, x) ∈ K 0 and condition 4 of the present theorem, it is easy to obtain that, ∀t ∈ [0, T],