On the existence of solution to multidimensional third order nonlinear equations

In this paper, we prove existence of an almost everywhere solution to mixed problem for a class of third order differential equations by non-zero rotation principle. Also studied the correctness of the formulation of the considered problem. 2010 Mathematics Subject Classifications: 35L76, 35L82.


Introduction
The paper is devoted to the problem of existence of almost everywhere solution and correctness of the formulation to the following multidimensional mixed problem for the third order nonlinear equation: where 0 < T < +∞; x = (x 1 , ..., x n ), Ω is a bounded n dimensional domain with an enough smooth boundary S; Γ = [0, T ] × S; functions a ij (x) (i, j = 1, n) and a(x) are measurable, and bounded in Ω and satisfy in Ω the following conditions: where ξ i (i = 1, 2, . . ., n) are arbitrary real numbers; ϕ, ψ are the given functions; F is some, generally speaking, nonlinear operator, and u(t, x) is a sought function.
Note that the results of this work improve the results of our article [1].There have been many works devoted to the study of mixed problems for nonlinear third order equations (see [2,3,5,8,10,12] and references therein), where the problem of existence and uniqueness in appropriate spaces, the problem of blow up of solutions and the problems of asymptotic behavior of solutions are studied.
As well as we know the equations considered in previous publications do not cover the class of equations we study.
Considered by us equations appear in modeling dynamical processes, in elasticity theory and in modeling dynamics of shallow water waves (see [7,11]).

Auxiliaries
In what follows we are using the following notations and facts.1.We denote by Ḋ(Ω) the class of all continuously differentiable functions on Ω which vanished near the boundary of Ω.The closure of Ḋ(Ω) with respect to the norm of W 1 2 (Ω) we denote by the class of all continuously differentiable functions on the cylinder Q T are equal to zero in the δ neighborhood of the lateral surface on the cylinder Q T , having the form: Q T,δ ≡ [0, T ] × Ω δ where Ω δ is a δ neighborhood of the boundary of Ω.The closure of Ḋ1 (Q T ) with respect to the norm of W 1 2 (Q T ) we denote by (1) almost everywhere in Q T and taking initial values (2) almost everywhere in Ω is called an almost everywhere solution of the problem (1)-(3).
2. For investigation of the problem (1)-( 3) we recall one property of the operator L, generating by the differential expression (4) and boundary condition (3): there are denumerable number of negative eigenvalues with the corresponding generalized eigenfunctions v s (x) which are complete and orthonormal in L 2 (Ω).We call function v s (x) ∈ • D(Ω) a generalized eigenfunction of the operator L, if it is not identically zero and for any function Φ(x) ∈ • D(Ω).As the system {v s (x)} ∞ s=1 is complete orthonormal in L 2 (Ω), then it is evident that every almost everywhere solution of problem ( 1)-(3) has the following form: where Then, after applying the Fourier method, finding the unknown Fourier coefficients u s (t) (s = 1, 2, ...) for the almost everywhere solution u(t, x) of the 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 every where solution of problem (1)-(3), it is easy to prove (see [1]) the following is any almost everywhere solution of problem (1)-( 3) and the generalized derivatives ∂ ∂x k a ij (x) (i, j, k = 1, 2, . . ., n) are bounded on Ω, then functions u s (t) (s = 1, 2, ...) satisfy system (5).
3. We denote by B α 0 ,...,α l β 0 ,...,β l ,T a totality of all the functions of the from We define the norm in this set as u = N T (u).
It is evident that all these spaces are Banach spaces ([6, p.50]). 4. Let G be class all functions u(t, x) which have the properties

On the existence of almost everywhere solution
In this section, using non-zero rotation principle, the following existence theorem for the almost everywhere solution of problem ( 1)-( 3) is proved for n: 3) ; the eigenfunctions v s (x) of the operator L under boundary condition v s (x)| S = 0 be three times continuously 2. F = F 1 + F 2 + F 3 , where a) the operator F 1 acts from the B 2 2,T into the space W 0,1 t,x,2 (Q T ) continuously and for all u ∈ B 2 2,T , t ∈ [0, T ] : where where M the boundary of the ball where Then problem (1)-( 3) has an almost everywhere solution.
Proof.Using condition 3 of this theorem, we have It is easy to obtain that, for any From (21) by virtue of the condition 2a this theorem it follows that the operator Q 1 acts continuously from the B 2 2,T into B 3,2 2,2,T .Since, the space B 3,2 2,2,T imbedded into the space B 2 2,T compactly ([6, Theorem 1.1, p.51]), then the operator Q 1 acts in the B 3,2 2,2,T compactly.
We consider in B 3,2 2,2,T the equations and a priori estimate their all the possible solutions u µ (t, x).Then, using inequality (6) ∀µ ∈ [0, 1] and t ∈ [0, T ] we have where C 0 is defined by (12).From (23), on applying Bellman's inequality [4, pp. 188,189] and using notations ( 8)- (10), we obtain that ∀µ ∈ [0, 1] : From here, using notation (7), we have that is, all the possible solutions u µ of equations ( 22) are a priori bounded in B 3,2 2,2,T and belong to the ball K 0 ( u B 3,2 2,2,T ≤ a 0 ).From ( 22) and (24) we obtain that ∀µ ∈ [0, 1] completely continuous vector field T µ = J − µQ 1 has no zeros on the boundary M of the ball K( u B 3,2 2,2,T ≤ a), where J is a unit vector field and a is a number appearing in the condition 2b this theorem.Consequently, completely continuous vector fields T 0 = J and T 1 = J − Q 1 are homotopic on the sphere M .Then their rotation δ on M are the same, namely: Now, we consider the operator Q 2 in the closed ball K. Just as the completely continuity of the operator Q 1 in B 3,2 2,2,T , was shown it is easy to show that the operator Q 2 acts compactly from K into B 3,2 2,2,T .Further, on the boundary M of the ball K we consider completely continuous vector fields Hence, in particular, it follows that completely continuous vector fields homotopic on the sphere M .Consequently, on M their rotations are equal to: And now in the ball K ρ ( u B 3,2 2,2,T ≤ ρ) we consider the operator Q 3 .Similar to (21), .
≤ ρ 0 ) (where the number ρ 0 is defined by ( 16)) and ε ∈ [0, 1] in the ball K ρ we consider the following equation Taking advantage fact that .
As q 0 < 1, then by virtue of the contracted mappings principle, the operator Due to notation (16) we have ( Consequently, for each ε ∈ [0, 1] the operator R ε is defined, in particular, and on (Q 1 + Q 2 )K.Due to (27) the operator R ε satisfies a Lipschitz condition (and therefore continuous) on (Q 1 + Q 2 )K.And as, the operators Q 1 and Q 2 are completely continuous on K, then for each ε ∈ [0, 1] the operator R ε (Q 1 + Q 2 ) compactly on K. Further, due to (17), for any u ∈ M and ε ∈ [0, 1] we have Consequently, the completely continuous vector field does not have zeros on M .Then does not have zeros on M same way the completely continuous vector field Thus, the completely continuous vector fields Hence, by virtue of the non-zero rotation principle [9, p. 207], the completely continuous vector field J − R 1 (Q 1 + Q 2 ) has at least one zero inside the ball of K. Since each such zero is a zero of the field J − Q 1 − Q 2 − Q 3 , it is thus proved that there exists in K at least one fixed point u(t, x) an operator Q 1 + Q 2 + Q 3 = Q.Further, it easy to verify (in absolutely the same way as in the proof of Theorem of [2]), that the function u(t, x) is an almost everywhere solution of problem (1)-(3).The theorem is proved.
(v) The operators F and F acts from where the function w(t, x) is defined by (11) and the number C 0 is defined by (12).Then for the unique almost everywhere solutions u(t, x) and ũ(t, x) of problems (1)-(3) and Ã, respectively, we have where the operator L is defined by ( 4) and the number C 0 is defined by (12).Proof: By Theorem 2 from work [1] each of the problems (1)-( 3) and Ã has a unique almost everywhere solution ,2,T .Then, by virtue of the lemma in section 2, the functions u s (t) (s = 1, 2, . ..) and ũs (t) (s = 1, 2, . ..) satisfy system (5), so that for ũs (t) (s = 1, 2, . ..) in the system (5) instead of ϕ s , ψ s and F(u) need to take φs , ψs and F(u), respectively.