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Gevrey solutions for equations of Euler-Bernoulli type

Authors
Journal
Journal of Differential Equations
0022-0396
Publisher
Elsevier
Publication Date
Volume
92
Issue
2
Identifiers
DOI: 10.1016/0022-0396(91)90053-c

Abstract

Abstract We consider solutions of an initial value problem involving the equation ∂ 2 ∂t 2 (x,t)+ ∑ i=0 4 a i(x) ∂ iw ∂x i (x,t) ∑ i=0 2 b i(x) ∂ i+1w ∂x i∂t (x,t)=0 for x, t > 0. Here, a 4( x) > 0 and is constant outside a compact set, b 2( x) ⩽ 0, and the coefficients a i ( x) for i ⩽ 3 and b i ( x) for i ⩽ 2 vanish outside a compact set. We show that the solutions are given by an integral operator, the kernel of which is an infinitely differentiable function of t for t > 0. In fact, its derivatives satisfy Gevrey-2 bounds.

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