Degeneration in differential equations with multiple characteristics

  • Kozhanov Aleksandr I., kozhanov@math.nsc.ru Sobolev Institute of Mathematics, 4 Koptyug Avenue, Novosibirsk 630090, Russia; Academy of Science of the Republic of Sakha (Yakutia), 33 Lenin Avenue, Yakutsk 677007, Russia
  • Lukina Galina A., lukina-g@mail.ru Ammosov North-Eastern Federal University, Mirny Polytechnic Institute, 5/1 Tikhonov Street, Mirny 678175, Russia
Keywords: differential equations with multiple characteristics, degeneration, boundary value problem, regular solution, existence, uniqueness

Abstract

We study the solvability of boundary value problems for the differential equations

$$\varphi(t)u_t+(-1)^m\psi(t)D^{2m+1}_{x}u+c(x,t)u=f(x,t),$$

$$\varphi(t)u_{tt}+(-1)^{m+1}\psi(t)D^{2m+1}_{x}u+c(x,t)u=f(x,t),$$

where $x\in(0, 1)$, $t\in(0, T),$ $m$ is a non-negative integer, $D^k_x=\frac{\partial^k}{\partial x^k}$ ($D^1_x=D_x$), while the functions $\varphi(t)$ and $\psi(t)$ are non-negative and vanish at some points of the segment $[0, T]$. We prove the existence and uniqueness theorems for the regular solutions, those having all generalized Sobolev derivatives required in the equation, in the inner subdomains.

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How to Cite
Kozhanov, A. and Lukina, G. (2021) “Degeneration in differential equations with multiple characteristics”, Mathematical notes of NEFU, 28(3), pp. 19-30. doi: https://doi.org/10.25587/SVFU.2021.91.97.002.
Section
Mathematics