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 Δu = 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→∞.