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Properties of Solutions of Equations Containing Powers of an Unbounded Operator

Properties of Solutions of Equations Containing Powers of an Unbounded Operator Differential Equations, Vol. 39, No. 10, 2003, pp. 1428–1439. Translated from Differentsial'nye Uravneniya, Vol. 39, No. 10, 2003, pp. 1355–1365. Original Russian Text Copyright c 2003 by Glushak. PARTIAL DIFFERENTIAL EQUATIONS Properties of Solutions of Equations Containing Powers of an Unbounded Operator A. V. Glushak Voronezh State Technical University, Voronezh, Russia Received July 12, 2001 The Cauchy problem 00 0 m+1 m u (t)+ (k=t)u (t)=(1) A u(t);t> 0; (1) u(0) = u;u (0) = 0; (2) for the Euler{Poisson{Darboux equation in a Banach space E wasconsideredin[1]. Here k  0, m 2 N, u 2 D (A ), and A is a linear closed operator such that the problem 00 0 0 u (t)+ (p=t)u (t)= Au(t);u(0) = u;u (0) = 0; (3) is uniformly well-posed for some p  0. We denote the set of such operators A by G and the resolving operator of problem (3) (the Bessel operator function) by Y (t;A). Thus if A 2 G , then problem (3) has a solution, which is unique p p and continuously depends on the initial data; moreover, u(t)= Y (t;A)u and p 0 kY (t;A)k M exp(!t);M  1;!  0: (4) p 0 0 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Differential Equations Springer Journals

Properties of Solutions of Equations Containing Powers of an Unbounded Operator

Glushak, A.
Differential Equations , Volume 39 (10) – Oct 11, 2004
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References (2)

Publisher
Springer Journals
Copyright
Copyright © 2003 by MAIK “Nauka/Interperiodica”
Subject
Mathematics; Difference and Functional Equations; Ordinary Differential Equations; Partial Differential Equations
ISSN
0012-2661
eISSN
1608-3083
DOI
10.1023/B:DIEQ.0000017916.89667.17
Publisher site
See Article on Publisher Site

Abstract

Differential Equations, Vol. 39, No. 10, 2003, pp. 1428–1439. Translated from Differentsial'nye Uravneniya, Vol. 39, No. 10, 2003, pp. 1355–1365. Original Russian Text Copyright c 2003 by Glushak. PARTIAL DIFFERENTIAL EQUATIONS Properties of Solutions of Equations Containing Powers of an Unbounded Operator A. V. Glushak Voronezh State Technical University, Voronezh, Russia Received July 12, 2001 The Cauchy problem 00 0 m+1 m u (t)+ (k=t)u (t)=(1) A u(t);t> 0; (1) u(0) = u;u (0) = 0; (2) for the Euler{Poisson{Darboux equation in a Banach space E wasconsideredin[1]. Here k  0, m 2 N, u 2 D (A ), and A is a linear closed operator such that the problem 00 0 0 u (t)+ (p=t)u (t)= Au(t);u(0) = u;u (0) = 0; (3) is uniformly well-posed for some p  0. We denote the set of such operators A by G and the resolving operator of problem (3) (the Bessel operator function) by Y (t;A). Thus if A 2 G , then problem (3) has a solution, which is unique p p and continuously depends on the initial data; moreover, u(t)= Y (t;A)u and p 0 kY (t;A)k M exp(!t);M  1;!  0: (4) p 0 0

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Differential Equations – Springer Journals

Published: Oct 11, 2004

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