Regular and singular perturbations of an abstract Euler-Poisson-Darboux equation. (English. Russian original) Zbl 0968.34041

Glushak, A. V.

Math. Notes 66, No. 3, 292-298 (1999); translation from Mat. Zametki 66, No. 3, 364-371 (1999).

From the text: Suppose that in the Banach space \(\mathbb{E}\) the Cauchy problem \[ u''(t)+ (k/t) u'(t)= \mathbb{A} u(t),\quad t>0,\tag{1} \] \[ u(0)= u_0,\quad u'(0)= 0,\tag{2} \] with a linear closed operator \(\mathbb{A}\) is uniformly well-posed. The author considers cases when equation (1) is perturbed by terms whose coefficients depend on \(t\), and he studies the behavior of the solution to problem (1), (2) as \(k\to 0\) (regular perturbation), as well as the singular perturbation if a parameter \(\varepsilon\to 0\) is introduced into the equation as a multiplier for \(u''(t)\).

MSC:

34G10

Linear differential equations in abstract spaces

Keywords:

regular and singular perturbations; Euler-Poisson-Darboux equation; linear closed operator

References:

[1]	A. V. Glushak, ”The operator Bessel function”,Dokl. Ross. Akad. Nauk [Russian Acad. Sci. Dokl. Math.],352, No. 5, 587–589 (1997). · Zbl 0965.34052
[2]	A. V. Glushak, ”Perturbations of an abstract Euler-Poisson-Darboux equation”,Math. Zametki, [Math. Notes],60, No. 3, 363–369 (1996). · Zbl 0898.34057
[3]	I. M. Gel’fand and B. M. Levitan, ”On the determination of a differential equation by its spectral function”,Izv. Akad. Nauk SSSR Ser. Mat. [Math. USSR-Izv.],15, No. 4, 309–360 (1951). · Zbl 0044.09301
[4]	V. Ya. Volk, ”On inversion formulas for a differential equation with singularity atx=0”,Uspekhi Mat. Nauk [Russian Math. Surveys],53, No. 4, 141–151 (1953). · Zbl 0053.24501
[5]	V. V. Gorodetskii, ”Representation of solutions to differential operator equations of hyperbolic type in polynomial form”,Izv. Vyssh. Uchebn. Zaved. Mat., [Soviet Math. (Iz. VUZ)], No. 11, 87–88 (1991).
[6]	V. V. Gorodetskii and M. L. Gorbachuk, ”Polynomial approximation of solutions of differential-operator equations in Hilbert space”,Ukrain. Mat. Zh. [Ukrainian Math. J.],36, No. 4, 500–502 (1984). · Zbl 0554.34042
[7]	A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev,Integrals and Series. Additional Chapters [in Russian], Nauka, Moscow (1986). · Zbl 0606.33001
[8]	G. H. Butcher and J. A. Donaldson, ”Regular and singular perturbation problems for a singular abstract Cauchy problem”,Duke Math. J.,42, No. 3, 435–445 (1975). · Zbl 0347.34043 · doi:10.1215/S0012-7094-75-04241-6
[9]	A. V. Glushak and V. D. Repnikov, ”Stabilization of solutions to a Cauchy problem for differential equations of first order in Banach space”,Dokl. Ross. Akad. Nauk [Russian Acad. Sci. Dokl. Math.],326, No. 2, 224–226 (1992). · Zbl 0796.34040