A solution to the Ricquier problem for a high-order parabolic equation in an unbounded domain: questions of existence and stabilization

The existence of a solution to the Ricquier problem with Dirichlet boundary conditions for one high-order parabolic equation in an unbounded domain is discussed. The convergence of the approximation using the Galerkin method of a generalized solution to the Ricquier problem is investigated. The dependence of behavior of the norm of the problem solution for large values of time on the geometry of an area unlimited in spatial variables lying at the base of the cylinder with a limited carrier of the initial function is considered. For a fairly wide class of unbounded domains, upper bounds for the solution of a mixed problem for a high-order parabolic equation are estimated, which are expressed in terms of simple geometric characteristics of the unbounded domain. The obtained upper estimations for the solution of the mixed problem are accurate.


Introduction
Most of the tasks of oil and gas industry are solved within the framework of continuum mechanics.Such problems are reduced to the systems of differential equations with several unknown functions, which represents the laws of mechanics.Initial-boundary value problems of a special kind are used to describe some physical processes.For example, a mathematical description of the gas flow with shock waves leads to an initial boundary value problem of a special kind, which we will call the Riquier problem.The concept of the "Riquier problem" is introduced for nonlinear partial differential equations based on the results of Sh.Riquier, N.M.Gunther, S.L. Sobolev and other mathematicians.The Riquier problem is a generalization of the Cauchy problem.The peculiarity of these tasks is that the initial conditions for some unknown functions are placed on one coordinate surface, and for the remaining oneson another one.For the first time such problem was observed by the French mathematician Sh .Riquier at the beginning of the last century.
A large number of papers have been devoted to the study of behavior at t → for solutions of parabolic equations [1,2].Those of them, in which questions are studied, in a certain sense close to the problem under consideration, are mainly devoted to parabolic equations of the second order, for which the maximum principle is known.In the works by A.K. Gushchin on the second mixed problem for a parabolic equation in an unbounded domain, a new direction of research was initiated.Later, these studies were continued in the works of V.I.Ushakov for the third mixed problem, A.V. Lezhnev for the second mixed problem, F.Kh. Mukminov, L.M. Kozhevnikova for the first mixed problem.High-order

Research methodology
The idea of obtaining off upper estimation of the solution in an unlimited domain is to establish the apriori estimation of the solution in the exterior of a ball of large radius, which ultimately reduces the problem to the upper estimation of the solution in a limited domain.The bottom estimation is based on the "uniformity" of the upper estimation, which is also valid for all bounded domains nested in an unbounded domain.This allows we to select such a limited domain for which the upper bound for the given cannot be improved.

Existence of a generalized solution
Let Ω be an arbitrary unbounded domain of a n-dimensional space n R , 2 n  , and Let us consider a following parabolic equation: where the initial function  has a bounded carrier lying in some ball satisfying the identity: x that coincides in the general domain of definition with the solution of the problem in T D for all 0. T  Theorem 1.If the boundary of the domain Ω belongs uniformly to class 2  C , then the problem ( 1)- ( 3) has a generalized solution 0,2 2,0 ( , ) ( ).
T u t x W D  Proof.We will carry out the proof using the Galerkin approximation method [6,7,8,9,10].Galerkin approximations have the form: Let us choose a sequence of functions () ). L  The functions i N C are determined from the conditions: 0, 0, 1, 2,..., .
we obtain the following system of N linear differential equations with N unknown functions ( ), 1, 2,..., .
The system (6) will have a unique solution if the initial conditions are given: .
Thus, the solution of problem ( 6)-( 7) is used and functions (4), called Galerkin approximations, are constructed from it.It is easy to show that they converge to the solution of the mixed problem (1)- (3).
It is not difficult to prove that if the N u is bounded in 0,2 2,0 ( ), ) .
Let's take advantage of the weak convergence of N u to u , LD , since this is a partial sum of the converging series.
Using limit in identity (10) we obtain identity:  The identity (11) is fulfilled for combinations of the form (9), and in determining the generalized solution, its validity is required for any function ). L  Now we assume that i  is orthonormal to 2 2,0 ( ).W  In fact, this can be done, since it is possible to move from the first assumption to the second and obtain from the basis i  of the space T WD It is enough to learn how to approximate == and v have a limited carrier, since such functions are dense in 1,2 2,0 () T WD.For such functions, the identity (11) will be true according to the Lemma: Lemma.For every function x == and the carrier v are bounded, there is a sequence Proof.For a fixed t , the function  can be decomposed according to the basis i  of the space 2 2,0 () . ii Due to the continuity of the scalar product, equality (13) can be differentiated by t :


According to the Lebesgue's theorem on bounded convergence, a limiting transition is possible under the sign of the integral.The lemma is proved.
Thus, the convergence of N vv → to 1,2 2,0 () T WD is proved, and hence the existence of a generalized solution of the mixed problem (1)-(3).

Estimation of the norm behavior of the task solution
In the study of the behavior of the norm of solution a mixed problem ( 1

0 R B of radius 0 R
. The requirement of the limitation of the carrier of the initial function is essential, became the rate of stabilization of the solution depends not only on the domain  , but also on the initial function  .A generalized solution of the problem (1) -

2 2
convergent subsequence can be distinguished from a bounded sequence.We can assume that N u weakly converge to u in 0,2 2,0 () T WD .Let N u weakly converge to , From the local compactness of the embedding of 0,

L
 the basis i  of the space 2 2,0 () W  by applying the Schmidt orthogonalization procedure, while the linear shells 12 , ,...,

-
be omitted.Problem (1)-(3) will be considered in the domain of the form: the radii of the largest balls ( , )Bz  centered in z , lying in []  ,