TY - JOUR
AU - Aleksandra Grigorieva
PY - 2022/02/01
Y2 - 2024/09/13
TI - The Dirichlet problem for the higher order composite type equations with discontinuous coefficients
JF - Mathematical notes of NEFU
JA - Math. notes NEFU
VL - 28
IS - 4
SE - Mathematics
DO - https://doi.org/10.25587/SVFU.2021.56.53.002
UR - https://mzsvfu.ru/index.php/mz/article/view/the-dirichlet-problem-for-the-higher-order-composite-type-equations
AB - 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).
ER -