Boundary value problems for a third order equation of mixed-composite type

We study the two boundary value problems for a third order equations of mixed-composite type. Under the certain conditions on the coefficients and the right-hand side of the equation, the regular solvability of these problems is proved. For both problems, the convergence estimate of the approximate solutions is obtained.


Introduction
Equations of mixed type are important in the applications to gas dynamics (see, for example, [1,2]). The well-posed formulation of the boundary value problem for mixed type equation with an arbitrary manifold of type switch was first proposed in [3]. The equations of composite type often arise in the mathematical models of real processes. The boundary value problems for such equations were studied in many publications (for example, [4,5]).
Here, a Vragov boundary value problem for a third order equation of mixed-composite type will be studied. The uniquely solvability of the problem with local boundary conditions will be proved, using the nonstationary Galerkin method and the regularization method.
An error estimate for approximate solutions of this problem in terms of the regularization parameter and the eigenvalues of the Dirichlet spectral problem for the Laplace operator will be obtained. Moreover, the boundary value problem with a integral boundary condition will be studied. Replacing the desired function, this problem will be reduced to previous boundary value problem, but for differential-integral equation. The regular solvability of this auxiliary problem will be proved, using the method of consecutive approximations. For the auxiliary problem and the nonlocal boundary problem, the convergence estimate of the approximate solutions will be obtained.

Formulation of the boundary value problems
Let Ω be a bounded domain in R n with a smooth boundary S. Denote Q = Ω × (0, T ), In the cylindrical domain Q consider the equation Assume that the coefficients of the equation (1) are sufficiently smooth. Introduce the sets  (1) in Q, such that Boundary value problem II. Find a solution of equation (1) in Q, such that the conditions (2), (4) are satisfied and where In an anisotropic Sobolev space W m,s 2 (Q) introduce the norm 3. Solvability of the boundary value problem I Introduce the class of functions Integration by parts proves the following assertion.
Then the following inequality holds: From Lemma 1 follows the uniqueness of a regular solution of the boundary value problem (1)- (4).
Approximate solution u N,ε (x, t) of the boundary value problem (1)-(4) has the form where c N,ε k (t) are defined as the solution of the following boundary value problem for the third order ordinary differential equation system: With the help of Lemma 1 we prove the following assertion.
Lemma 2 Assume the conditions of Lemma 1 hold, f ∈ L 2 (Q) and the function k(x,t) satisfies either of the conditions: Then for an approximate solution u N,ε (x, t) the following estimate holds: From Lemma 2 follows the uniquely solvability of the boundary value problem (6), (7 p ), p = 1, 2.
We prove a priory estimates for approximate solutions u N,ε (x, t) and, correspondingly, the following theorem.
Theorem 2 Assume all conditions of Theorem 1 are satisfied. Then the following error estimate for u N,ε (x, t) holds: where u(x, t) is the exact solution of the boundary value problem (1)-(4).
If the function u(x, t) ∈ W L is a solution of the nonlocal boundary value problem (1), (2), (4), (5) then the function will be a solution for the following problem.
Auxiliary problem. Find a solution of equation such that the boundary conditions (2)-(4) are satisfied, where Due to the Theorem 1, with N (t) ≡ 0 the boundary value problem (8), (2)-(4) has a unique solution v 0 (x, t) ∈ W L and the following estimate holds: For all funtions v(x, t) ∈ W L the following inequality holds: We prove the following theorem, using the method of consecutive approximations with v 0 (x, t) as initial approximation.
Theorem 3 Assume all conditions of Theorem 1 are satisfied and Then the boundary value problem (8), (2)-(4) has a unique solution v(x, t) ∈ W L and the convergence estimate holds: where v m (x, t) is an approximation with number m for v(x, t).

Solvability of the boundary value problem II
It is proved above that the boundary value problem (8), (2)-(4) has a unique solution v(x, t) ∈ W L . Denote Then it is easy to prove the following assertion.