Nonlocal Boundary-Value Problem for a Third-Order Parabolic-Hyperbolic Equation with Degeneration of Type and Order in t
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NONLOCAL BOUNDARY-VALUE PROBLEM FOR A THIRD-ORDER PARABOLIC-HYPERBOLIC EQUATION WITH DEGENERATION OF TYPE AND ORDER IN THE HYPERBOLICITY DOMAIN Zh. A. Balkizov
UDC 517.956.6
Abstract. We consider a nonlocal boundary-value problem for a third-order parabolic-hyperbolic equation with degeneracy of type and order in the domain of hyperbolicity, containing second-order derivatives in the boundary conditions. Sufficient conditions of the unique solvability of the problem are obtained. The Tricomi method is used to prove the uniqueness theorem for a solution. The solution of the problem is expressed in the explicit form. Keywords and phrases: degenerate hyperbolic equation of the first kind, equation with multiple characteristics, third-order parabolic-hyperbolic equation, mixed boundary-value problem, nonlocal boundary-value problem, Tricomi problem, Tricomi method, Volterra integral equation of the second kind, Fredholm integral equation of the second kind. AMS Subject Classification: 35M10, 35M13
1.
Introduction. On the Euclidean plane of the points (x, y), we consider the equation (−y)m uxx − uyy + a(−y)(m−2)/2 ux , y < 0, 0= uxxx − uy + bux , y > 0,
(1)
where u = u(x, y) is the unknown function; m, a, b = const, m > 0, and |a| ≤ m/2. Equation (1) for y < 0 coincides with the degenerate hyperbolic equation of the first kind (see [35]) (−y)m uxx − uyy + a(−y)(m−2)/2 ux = 0,
(2)
and for y > 0 ”— with the third-order equation with multiple characteristics (see [13]) uxxx − uy + bux = 0.
(3)
Equation (2) is a hyperbolic equation with parabolic degeneration of the first kind along the line y = 0. For m = 2, Eq. (2) turns into the Bitsadze–Lykova equation (see [10, 24, 26]), and for a = 0, it becomes the Gellerstedt equation, which, as was shown in [27], can be used in the problem of determining the shape of dam slots. A special case of Eq. (2) is also the Tricomi equation, which is the theoretical basis of nearsonic gas dynamics (see [9, 16]). In [35], a solution of the Cauchy problem for Eq. (2) for |a| ≤ m/2 was described. The study of the first and second Darboux problems for Eq. (2) is described in [18]. In [19], a continuity criterion for solutions of the Goursat problem for a degenerate hyperbolic equation of the form (2) was considered. Equation (3) coincides with the linearized Korteweg–de Vries equation (see [22]), which is used in important applications in problems of propagation of nonlinear waves in weakly dispersive media (see [20]). According to the classification given in [26], Eq. (3) refers to parabolic equations, whereas in [13] such equations are called third-order equations with multiple characteristics. In [12], based on methods of the theory of potentials and integral Laplace transform, boundary-value problems for Eq. (3) were studied; this problem is now called the Cattabriga problem. By using fundamental solutions obtained in [12, 13], the Green function of the Cattabriga problem for Eq. (3) was constructed, Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheni
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