Boundary-Value Problems for Ultraparabolic and Quasi-Ultraparabolic Equations with Alternating Direction of Evolution
- PDF / 206,329 Bytes
- 8 Pages / 594 x 792 pts Page_size
- 50 Downloads / 190 Views
BOUNDARY-VALUE PROBLEMS FOR ULTRAPARABOLIC AND QUASI-ULTRAPARABOLIC EQUATIONS WITH ALTERNATING DIRECTION OF EVOLUTION A. I. Kozhanov
UDC 517.946
Abstract. We examine the solvability of boundary-value problems for the differential equation h(t)ut + (−1)m Da2m+1 u − Δu + c(x, t, a)u = f (x, t, a); ∂k , ∂ak where the sign of the function h(t) arbitrarily alternates in the interval [0, T ]. The existence and uniqueness theorems of regular (i.e., possessing all generalized derivatives in the Sobolev sense) solutions are proved. x ∈ Ω ⊂ Rn ,
0 < t < T,
0 < a < A,
Dak =
Keywords and phrases: ultraparabolic equation, odd-order nonclassical differential equation with alternating direction of evolution, boundary-value problem, regular solution, existence, uniqueness. AMS Subject Classification: 35M99, 35K70
1. Introduction. In [4], the solvability of boundary-value problems for some differential equations with two time variables and alternating direction of evolution was studied, more precisely, the equations ∂ 2m+1 u − uxx + c(x, t, a)u = f (x, t, a), (1) ∂a2m+1 where m is a nonnegative integer and h(x) is such that xh(x) > 0 for x = 0. For these equations, theorems on the existence and uniqueness of regular (i.e., possessing all Sobolev generalized derivatives involved in the equation) solutions of the Gevrey problem (see [3]) and the Fichera problems (see [2, 6]) were proved. In this paper, we study equations similar to Eq. (1) with a function h depending on the variables t and x, where x runs over a bounded domain in Rn . Let us also specify that the change of the direction of evolution in the equations studied can occur not once (as in Eq. (1)) but arbitrarily many times and that the function h(t) can vanish on a set of positive measure. Before proceeding to the substantive part of the paper, we note the following. First, for ultraparabolic equations (for m = 0) with alternating direction of evolution, the solvability of boundaryvalue problems was studied in [7] for model situations with another character of alternating direction of evolution than described in the present work. Second, quasiultraparabolic equations (Eq. (1) in the case m > 0) with alternating direction of evolution determined by the function h(t) have not been studied earlier. h(x)ut + (−1)m
2. Statement of problems. Let Ω be a bounded domain in Rn with a smooth (for simplicity, infinitely differentiable) boundary Γ, T and A be given positive numbers, and Q be a cylinder of the space Rn+2 with the variables (x, t, a): Q = Ω × (0, T ) × (0, A). Next, let h(t), c(x, t, a), and f (x, t, a) be given functions defined for x ∈ Ω, t ∈ [0, T ], and a ∈ [0, A], and let m be a given nonnegative integer. For a nonnegative integer k, we denote by Dak the derivative ∂ k /∂ak . Finally, we denote by L the differential operator acting on a given function v(x, t, a) by the rule Lv = h(t)vt + (−1)m Da2m+1 v − Δv + c(x, t, a)v. Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 149, Proceedings of the In
Data Loading...
