On the possible rate of decay at infinity of solutions of second order partial differential equations

For second-order partial differential equations the question of whether they can have solutions decaying superexponentially at infinity is studied. An example is constructed of an equation $\Delta u=q(x)u$ on the plane with bounded coefficients $q$ having a nonzero solution decaying superexponentially. This example provides a negative answer to a familiar question of E. M. Landis. These questions are also studied for hyperbolic and parabolic equations on manifolds. An example is constructed of a parabolic equation having a nonzero solution $u(x,t)$ decaying superexponentially as $t\to\infty$.