Boundary-Value Problem for a Third-Order Hyperbolic Equation that is Degenerate Inside a Domain and Contains the Aller O
- PDF / 199,450 Bytes
- 8 Pages / 594 x 792 pts Page_size
- 26 Downloads / 169 Views
BOUNDARY-VALUE PROBLEM FOR A THIRD-ORDER HYPERBOLIC EQUATION THAT IS DEGENERATE INSIDE A DOMAIN AND CONTAINS THE ALLER OPERATOR IN THE PRINCIPAL PART R. Kh. Makaova
UDC 517.95
Abstract. Boundary-value problems for a third-order hyperbolic equation that is degenerate inside a domain and contains the Aller operator in the principal part, are examined. The existence and uniqueness theorem for solutions of the problem is proved. Keywords and phrases: boundary-value problem, third-order hyperbolic equation, Aller equation. AMS Subject Classification: 35L25, 35L80
1. Introduction. In the Euclidean plane with the coordinates (x, y), we consider the following equation: uy − auxx − buxxy , y > 0, (1) 0= m−2 m (−y) uxx − uyy − c(−y) 2 ux , y < 0, where a, b, and m are given positive numbers, |c| ≤ m/2, and u = u(x, y) is the unknown real-valued function of the independent variables x and y. Equation (1) coincides with the Aller equation (see [8]) for y > 0: ∂ ∂u ∂2u ∂u = a +b , (2) ∂y ∂x ∂x ∂x∂y and with the degenerate hyperbolic equation of the first kind (see [18]) for y < 0: (−y)m uxx − uyy − c(−y)
y B0
Br
T Ω+ A0 0
Ar r x
r 2
Ω−
m−2 2
ux = 0.
(3)
Let Ω+ = {(x, y) : 0 < x < r, 0 < y < T }. We denote by Ω− the domain bounded by the characteristics of Eq. (3): m+2 2 (−y) 2 = 0, A0 C : x − m+2 m+2 2 (−y) 2 = r, Ar C : x + m+2 which leave the points A0 = (0, 0), Ar = (r, 0) and intersect each other at the point C = r/2, − [(m + 2)r/4]2/(m+2) and A0 Ar = {(x, 0) : 0 < x < r}. Let B0 = (0, T ), Br = (r, T ) and A0 B0 = {(0, y) : 0 < y < T },
Ar Br = {(r, y) : 0 < y < T }, Ω = Ω+ ∪ Ω− ∪ (A0 Ar ). It is known (see [4]) that under certain assumptions, Eq. (2) describes the motion of soil moisture, and its solution is interpreted as soil moisture with the diffusivity coefficient a and the coefficient of C
Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 149, Proceedings of the International Conference “Actual Problems of Applied Mathematics and Physics,” Kabardino-Balkaria, Nalchik, May 17–21, 2017, 2018.
780
c 2020 Springer Science+Business Media, LLC 1072–3374/20/2505–0780
moisture conductivity b at the point x of the soil layer 0 ≤ x ≤ r at the point of time y ∈ [0, T ] (see [13]). Although Eq. (2) is an equation, some researchers refer to it as a pseudo-parabolic equation (see [17]). The works [3, 5, 6, 16, 17, 19] are devoted to the solution of various local, nonlocal, and mixed boundary-value problems for third-order pseudo-parabolic equations. Equation (3) is a hyperbolic equation, and for m = 2 it is called the Bitsadze–Lykov equation (see [14]). For c = 0, Eq. (3) turns into the Gellerstedt equation, which is used in the problem on the optimal shape of the dam slot (see [12]). In [9, 10], the first and second Darboux problems for Eq. (3) were studied. The works [1, 2, 18] are devoted to the study of various boundary-value problems for degenerate hyperbolic equations. In this paper, we pose and examine question on the unique solvabili
Data Loading...
