We consider the annulus \begin{align*} A = B_4(0,0)\setminus B_1(0,0) \end{align*} in $\mathbb{R}^2$. And let $J$ be the segment in $A$ \begin{align*} J = \{(x,y) \in A : 2<x<3, y=0 \}. \end{align*}
Can someone help me figure out whether the following Cauchy problem is ill-posed? And if so, why?
\begin{align}\label{cauchy prob} \begin{cases} z_t-\Delta z = 0& \qquad \text{on $A \times (0,T)$;} \\ z(x,0,t) = 0& \qquad \text{on $J\times (0,T)$;}\\ z_y(x,0,t) = 0& \qquad \text{on $J\times (0,T)$.}\\ \end{cases} \end{align}
Thank you in advance