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 Lq with q∈ [2 , ∞) , 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 Lq. As a corollary we show also the Liouville-type theorem for both forward and ancient solutions.
- Asymptotic behavior
- Elliptic and parabolic equations of second order
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