Boundary value problems for twice degenerate differential equations with multiple characteristics

  • Kozhanov Alexander I., kozhanov@math.nsc.ru Sobolev Institute of Mathematics, 4 Acad. Koptyug Avenue, Novosibirsk 630090, Russia
  • Zikirov Obidjon S., zikirov@yandex.com Faculty of Mechanics and Mathematics, National University of Uzbekistan, 4 University Street, Vuzgorodok, Tashkent 100174, Uzbekistan
Keywords: differential equations of odd order, degeneracy, change of direction of evolution, boundary value problems, regular solutions, existence, uniqueness

Abstract

We study the solvability of boundary value problems for degenerate differential equations of the form $$\varphi(t)u_t-(-1)^m\psi(t)D^{2m+1}_{x}u+c(x,t)u=f(x,t)$$ ($D^k_x=\frac{\partial^k}{\partial x^k},\,m\geq 1$ is an integer, $x\in(0,1),\, t\in(0,T),\,0<T<+\infty),$ called equations with multiple characteristics. In these equations, the function $\varphi(t)$ can change the sign on the interval $[0, T]$ arbitrarily, while the function $\psi(t)$ is assumed nonnegative. For the equations under consideration, we propose the formulation of boundary value problems which are essentially determined by numbers $\varphi(0)$ and $\psi(T)$. Existence and uniqueness theorems are proved for the regular solutions that have all Sobolev generalized derivatives entering into the equation.

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How to Cite
Kozhanov, A. and Zikirov, O. (&nbsp;) “Boundary value problems for twice degenerate differential equations with multiple characteristics”, Mathematical notes of NEFU, 25(4), pp. 34-44. doi: https://doi.org/10.25587/SVFU.2018.100.20552.
Section
Mathematics