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BOUNDARY-VALUE PROBLEM FOR A LOADED HYPERBOLIC-PARABOLIC EQUATION WITH DEGENERACY OF ORDER IN THE HYPERBOLICITY DOMAIN K. U. Khubiev

UDC 517.95

Abstract. In this paper, we study a boundary-value problem for a loaded hyperbolic-parabolic equation with degeneracy of order in the domain of its hyperbolicity. The existence and uniqueness theorems for solutions of this problem are proved and a representation of solutions is obtained. Keywords and phrases: loaded equation, mixed-type equation, hyperbolic equation, parabolic equation, boundary-value problem. AMS Subject Classification: 35M10, 35M12

1.

Introduction and statement of the problem. Consider the equation  uxx − uy + λ1 u(x1 , y) = 0, y > 0, ux + uy + cu + λ2 u(x2 , y) = 0,

y < 0,

(1)

in the domain Ω bounded by the segments AA1 , BB1 , and A1 B1 of lines x = 0, x = r, and y = T > 0, respectively, for y > 0; by the characteristics AA2 : x − y = 0 and BB2 : x − y = r of Eq. (1), and the segment A2 B2 of the line y = x2 − r for y < 0; here x1 ∈ [0, r], x2 ∈ [0, r[, and λ1 , λ2 , and c are constants. Loaded equations of mixed type play an important role in the theory of heat and mass transfer in composite media with fractal organization and memory (see [15]). Numerous problems related to forecasting and regulation of the groundwater level (see [15, p. 95]) are reduced to boundary-value problems for equations of the form (1) for y > 0, as well as the problems describing the process of heat propagation in a one-dimensional bounded medium containing a heat source whose power is proportional to the temperature (see [6]). For y < 0, Eq. (1) is a loaded one-velocity transport equation; in mathematical biology for λ2 = 0 it is known as the McKendrick equation (see [14, p. 179]). The equation m  ci (x, y)u(xi , y) = 0, ux + uy + c(x, y)u + i=1

whose particular case is Eq. (1) for y < 0, describes the dynamics of a closed population in the case where the number of organisms of the ages x1 , x2 , . . . , xm at each moment of time y varies (see [14, p. 244]). In [9], Eq. (1) for y < 0 describes the dynamics of the age structure of a limited population. Boundary-value problems for loaded partial differential equations were studied in numerous works (see, e.g., [7, 15] and the references therein). At present, the theory of boundary-value problems for mixed-type loaded equations is being intensively developed by many scientists. Boundary-value problems for model loaded equations of mixed hyperbolic-parabolic and elliptic-hyperbolic type with loading term λu(x, 0) and degeneracy of order in the domain of their hyperbolicity in bounded and unbounded domains were investigated in [15]. Also, inner boundary-value problems for mixed hyperbolic-parabolic equations with characteristic and noncharacteristic loads and their connection with analogs of the Tricomi problem were also considered (see Nagrur). In [2], the influence of the “load” λux (x, 0) + μuy (0, y) on the well-posedness of initial-boundary value problems for essentially Translated from Itogi Nauki i Tekhniki, Seriya