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

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