  We study an asymptotic behavior of solutions to elliptic equations of the
second order in a two dimensional exterior domain. Under the assumption that
the solution belongs to $L^q$ with $q \in [2,\infty)$, we prove a pointwise
asymptotic estimate of the solution at the spatial infinity in terms of the
behavior of the coefficients. As a corollary, we obtain the Liouville-type
theorem in the case when the coefficients may grow at the spacial infinity. We
also study a corresponding parabolic problem in the $n$-dimensional whole space
and discuss the energy identity for solutions in $L^q$. As a corollary we show
also the Liouville-type theorem for both forward and ancient solutions.
