Second-Order Abstract Cauchy Problem in Two Variables

Abstract

In this paper, we consider the following second-order abstract Cauchy
problem in two variables
\begin{equation*}
D_{ss}^{2}u(s,t)+D_{tt}^{2}u(s,t)+2D_{st}^{2}u(s,t)+A\left[
D_{s}u(s,t)+D_{t}u(s,t)\right] =Bu(s,t),
\end{equation*}
where $A,B$ are closed linear operators on a Banach space $X$ such that
\begin{eqnarray*}
A &:&Dom(A)\subseteq X\rightarrow X, \\
B &:&Dom(B)\subseteq X\rightarrow X,
\end{eqnarray*}
$u(s,t):\left[ 0,1\right] \times \left[ 0,1\right] \rightarrow X$ is an
unknown twice-continuously partially differentiable function on $\left[ 0,1%
\right] \times \left[ 0,1\right] \subseteq 

\mathbb{R}^{2}$ where $Rang(u)\subseteq Dom(A)\cap Dom(B)$. Utilizing properties of atomic operators, an atomic solution is obtained.