Boundary-Value Problems for Ultraparabolic and Quasi-Ultraparabolic Equations with Alternating Direction of Evolution

Abstract

We examine the solvability of boundary-value problems for the differential equation

$$ h(t){u}_t+{\left(-1\right)}^m{D}_a^{2m+1}u-\varDelta u+c\left(x,t,a\right)u=f\left(x,t,a\right); $$

$$ {\displaystyle \begin{array}{ccc}x\in \Omega \subset {\mathrm{\mathbb{R}}}^n,& 0<t<T,& 0<a<A\end{array}},\kern0.5em {D}_a^k=\frac{\partial^k}{\partial {a}^k}, $$

where the sign of the function h(t) arbitrarily alternates in the interval [0, T]. The existence and uniqueness theorems of regular (i.e., possessing all generalized derivatives in the Sobolev sense) solutions are proved.