Nontrivial ordered ω-limit sets in a linear degenerate parabolic equation

$ u_t=a(x) (\Delta u+\lambda_1 u) \qquad $ (*)

with zero Dirichlet data in a smoothly bounded domain $\Omega \subset \R^n$, $n\ge 1$. Here $a$ is positive in $\Omega$ and Hölder continuous in $\bar\Omega$, and $\lambda_1>0$ denotes the principal eigenvalue of $-\Delta$ in $\Omega$ with Dirichlet data. It is shown that if $\int_\Omega \frac{(\dist(x,\partial\Omega))^2}{a(x)}dx=\infty$ then there exist initial data in $W^{1,\infty}(\Omega)$ such that the solution of (*) is bounded but not convergent as $t\to\infty$: It has a totally ordered $\omega$-limit set which is not a singleton. Under the above condition, the occurrence of even unbounded ordered $\omega$-limit sets is demonstrated. Conversely, if $\frac{(\dist(x,\partial\Omega))^2}{a(x)}$ is integrable then any solution emanating from initial data in $W^{1,\infty}(\Omega)$ converges to some stationary solution of (*) as time approaches infinity.