Задача Дирихле для уравнений составного типа высокого порядка с разрывными коэффициентами

Aleksandra I. Grigorieva

doi:10.25587/SVFU.2021.56.53.002

Grigorieva Aleksandra I., shadrina_ai@mail.ru Ammosov North-Eastern Federal University, Institute of Mathematics and Informatics, 48 Kulakovsky Street, Yakutsk 677000, Russia; Academy of Sciences of the Republic of Sakha (Yakutia), 33 Lenin Avenue, Yakutsk 677000, Russia

DOI: https://doi.org/10.25587/SVFU.2021.56.53.002

Keywords: differential composite type equations, the Dirichlet problem, blow-up coefficient, regular solution, existence, uniqueness

Abstract

We study the Dirichlet problem for the composite type differential equations

$$D_t\big[(-1)^pD^{2p+1}_tu-h(x)u_{xx}\big]+a(x)u_{xx}+c(x,t)u=f(x,t)$$

in the domain $Q=\{(x,t)\,:\,x\in(-1,0)\cup(0,1),\,t\in(0,T),\,0<T<+\infty\}$, where

$p \geq 1$ is an integer, $D^k_t=\frac{\partial^k}{\partial t^k},$ and $D_t=\frac{\partial}{\partial t}$. The feature of such equations is that the coefficients $h(x)$ and $a(x)$ can have a discontinuity of the first kind when passing through the point $x = 0$. In addition to the usual Dirichlet boundary conditions, the problem under study also specifies the conjugation conditions on the line $x = 0$. Existence and uniqueness theorems are proved for regular solutions (those having all generalized Sobolev derivatives).

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The Dirichlet problem for the higher order composite type equations with discontinuous coefficients

Abstract

References