We prove that a matrix-differential operator A in a Banach space is a Fredholm operator with zero index and construct the corresponding subspaces and projections. We find the spectrum of A. Based on properties of the operator A, we study solutions to the initial-boundary value problem for a second order partial differential equation with a small parameter ε at the higher order derivatives and some parameter c at lower order terms. We clarify how the parameter c affects the behavior of the solution as ε → 0.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
S. M. Nikol’skii, “Linear equations in linear normed spaces” [in Russian], Izv. AN SSSR, Ser. mat. 7, No. 3, 147–166 (1943).
S. Zubova and K. Chernyshov, “On the linear differential equation with a Fredholm operator at a derivative” [in Russian], Differ. Uravn. Primen. 14, 21–39 (1976).
S. P. Zubova, “Solution of the homogeneous Cauchy problem for an equation with a Fredholm operator multiplying the derivative,” Dokl. Math. 80, No. 2. 710–712 (2009).
S. P. Zubova, On the solvability of the Cauchy problem for the description quasiregular equations in Banach spaces” [in Russian], Vestn. Voronezh. Gos. Univ., Ser. Fiz. Mat. No. 2, 192-198 (2013).
S. P. Zubova and E. V. Raetskaya, “Solution of the Cauchy problem for two descriptive equations with Fredholm operator,” Dokl. Math. 90, No. 3, 732–736 (2014).
S. G. Krein and N. Z. Kan, “Asymptotic method in the problem of oscillations of a strongly viscous fluid,” J. Appl. Math. Mech. 33, No. 3, 442–450 (1969).
S. P. Zubova, “The role of perturbations in the Cauchy problem for equations with a Fredholm operator multiplying the derivative,” Dokl. Math. 89, No. 1, 72–75 (2014).
M. M. Vainberg and V. A. Trenogin, Theory of Branching of Solutions of Nonlinear Equations Nordhoff, The Netherlands (1974).
V. A. Trenogin, “The development and applications of the asymptotic method of Lyusternik and Vishik,” Russ. Math. Surv. 25, No. 4, 119–156 (1970).
S. P. Zubova and V. I. Uskov, “The asymptotic solution of a singularly perturbed Cauchy problem for the first order equation in a Banach space” [in Russian], Vestn. Voronezh. Gos. Univ., Ser. Fiz. Mat. No. 3, 147–155 (2016).
S. A. Lomov and I. S. Lomov, Fundamentals of the Mathematical Boundary Layer Theory [in Russian], Moscow State Univ. Press, Moscow (2011).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Problemy Matematicheskogo Analiza 109, 2021, pp. 97-108.
Rights and permissions
About this article
Cite this article
Zubova, S.P., Raetskaya, E.V. & Uskov, V.I. Degeneracy Property of a Matrixdifferential Operator and Applications. J Math Sci 255, 640–652 (2021). https://doi.org/10.1007/s10958-021-05401-7
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-021-05401-7