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Uniqueness in a doubly characteristic Cauchy problem. (English) Zbl 0692.35066
The main background for the investigations of this paper is given by a paper of F. Treves [Proc. Am. Math. Soc. 46, 229-233 (1974; Zbl 0297.35048)] and a paper of B. Birkeland and J. Persson [J. Differ. Equations 30, 64-88 (1978; Zbl 0361.35039)]. The authors start with a discussion of Carleman estimates and a corresponding uniqueness result for differential equations \[ (\partial /\partial x-a(x)\partial /\partial y)(\partial /\partial x+a(x)\partial /\partial y)u+b(x)\partial u/\partial y=0. \] Then they specialize to the equation \[ (\partial /\partial x+ax^ k\partial /\partial y)(\partial /\partial x-ax^ k\partial /\partial y)u-cx^{k-1}\partial u/\partial y=0 \] and discuss uniqueness in the class \(C^ m\). In the last section they give an extension of results of Birkeland and Persson to differential equations \[ u_{xx}-a(x)u_{xy}-b(x)u_{yy}-c(x)u_ x-d(x)u_ y-e(x)u=0. \] They show how uniqueness in the class of distributions can be obtained from uniqueness in the class \(C^ m\), for some \(m\geq 2\).
Reviewer: W.Watzlawek
MSC:
35L80 Degenerate hyperbolic equations
35L15 Initial value problems for second-order hyperbolic equations
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