Well-Posedness of a Mixed Problem in a Cylindrical Domain for One Class of Multidimensional Hyperbolic-Parabolic Equatio
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BRIEF COMMUNICATIONS WELL-POSEDNESS OF A MIXED PROBLEM IN A CYLINDRICAL DOMAIN FOR ONE CLASS OF MULTIDIMENSIONAL HYPERBOLIC-PARABOLIC EQUATIONS S. A. Aldashev
UDC 517.956
We prove the unique solvability and obtain the explicit expression for the classical solution of mixed problem in a cylindrical domain for one class of multidimensional hyperbolic-parabolic equations.
Introduction Mixed problems in cylindrical domains for multidimensional hyperbolic equations in generalized spaces were investigated in [1, 2]. In [3], the well-posedness of the mixed problem was proved and an explicit expression for its classical solution was obtained. To the best of our knowledge, for multidimensional hyperbolic-parabolic equations, these problems are not studied. In the present paper, we prove the unique solvability and deduce explicit expression for the classical solution of the mixed problem for a class of multidimensional hyperbolic-parabolic equations. 1. Statement of the Problem Let ⌦↵β be a cylindrical domain of the Euclidean space Em+1 of points (x1 , . . . , xm , t) bounded by a cylinder Γ = {(x, t) : t|x| = 1} and two planes t = ↵ > 0 and t = β < 0, where |x| is the length of the vector x = (x1 , . . . , xm ). By ⌦↵ and ⌦β we denote parts of the domain ⌦↵β and by Γ↵ and Γβ we denote parts of the surface Γ lying in the half spaces t > 0 and t < 0. Also let σ↵ and σβ be the upper and lower bases of the domain ⌦↵β , respectively. Further, let S be the common part of the boundaries of the domains ⌦↵ and ⌦β , i.e., the set {t = 0, 0 < |x| < 1} in Em . In the domain ⌦↵β , we consider the following multidimensional hyperbolic-parabolic equations:
0=
8 m X > > ∆x u − utt + ai (x, t)uxi + b(x, t)ut + c(x, t)u, > > < > > > > :∆x u − ut +
i=1 m X
di (x, t)uxi + e(x, t)u,
t > 0, (1) t < 0,
i=1
where ∆x is the Laplace operator with respect to the variables x1 , . . . , xm , m ≥ 2. Abai Kazakhstan National Pedagogic University, Almaty, Kazakhstan; e-mail: e -mail: [email protected]. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, No. 2, pp. 280–288, February, 2020. Original article submitted November 10, 2017. 0041-5995/20/7202–0317
© 2020
Springer Science+Business Media, LLC
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S. A. A LDASHEV
318
In what follows, it is convenient to pass from Cartesian coordinates x1 , . . . , xm , t to spherical coordinates r, ✓1 , . . . , ✓m−1 , t, r ≥ 0, 0 ✓i ⇡, i = 2, 3, . . . , m − 1, 0 ✓1 < 2⇡, ✓ = (✓1 , . . . , ✓m−1 ). Problem 1. To find the solution of Eq. (1) in the domain ⌦↵β for t 6= 0 from the class ¯ ↵ ) \ C 2 (⌦↵ [ ⌦β ) ¯ ↵β ) \ C 1 (⌦↵β ) \ C 1 (⌦ C(⌦ satisfying the boundary conditions
� u �Γ = β
and, moreover,
1 (0, ✓)
=
2 (0, ✓)
and
� u � Γ↵ =
2 (t, ✓),
2 (β, ✓)
(2)
1 (t, ✓),
� u �σ = '(r, ✓)
(3)
β
= '(1, ✓).
� k Let Yn,m (✓) be a system of linearly independent spherical functions of order n, 1 k kn , let (m − 2)!n!kn = (n + m − 3)!(2n + m − 2),
and let W2l (S), l = 0, 1, . . . , be Sobolev spaces. The following lemma is true [4]: Lemma 1. Suppose that a function f (r, ✓) belong
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