Wiener Tauberian theorem and half-space problems for parabolic and elliptic equations

For various kinds of parabolic and elliptic partial differential and differential-difference equations, results on the stabilization of solutions are presented. For the Cauchy problem for parabolic equations, the stabilization is treated as the existence of a limit as the time unboundedly increases. For the half-space Dirichlet problem for parabolic equations, the stabilization is treated as the existence of a limit as the independent variable orthogonal to the boundary half-plane unboundedly increases. In the classical case of the heat equation, the necessary and sufficient condition of the stabilization consists of the existence of the limit of mean values of the initial-value (boundary-value) function over balls as the ball radius tends to infinity. For all linear problems considered in the present paper, this property is preserved (including elliptic equations and differential-difference equations). The Wiener Tauberian theorem is used to establish this property. To investigate the differential-difference case, we use the fact that translation operators are Fourier multipliers (as well as differential operators), which allows one to use a standard Gel'fand-Shilov operational scheme. For all quasilinear problems considered in the present paper, the mean value from the stabilization criterion is changed: It undergoes a monotonic map, which is explicitly constructed for each investigated nonlinear boundary-value problem.