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

A. I. Grigorieva

Matematičeskie zametki SVFU, Tome 28 (2021), pp. 17-29.

Résumé

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 $Q=\{(x,t)\,:\,x\in(-1,0)\cup(0,1),\,t\in(0,T),\,0$, 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).

Détail

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