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Mathematics
The dirichlet problem for the higher order composite type equations with discontinuous coefficients
A. I. Grigorievaab a Ammosov North-Eastern Federal University,
Institute of Mathematics and Informatics,
48 Kulakovsky Street, Yakutsk 677000, Russia
b Academy of Sciences of the Republic of Sakha (Yakutia),
33 Lenin Avenue, Yakutsk 677000, Russia
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).
Keywords:
differential composite type equations, the Dirichlet problem, blow-up coefficient, regular solution, existence, uniqueness.
Received: 20.10.2021 Revised: 20.10.2021 Accepted: 26.11.2021
Citation:
A. I. Grigorieva, “The dirichlet problem for the higher order composite type equations with discontinuous coefficients”, Mathematical notes of NEFU, 28:4 (2021), 17–29
Linking options:
https://www.mathnet.ru/eng/svfu331 https://www.mathnet.ru/eng/svfu/v28/i4/p17
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