Nonclassical dynamical behavior of solutions of partial differential-difference equations

For partial differential-difference equations, a review of results regarding the relation between the type of the equation and dynamical properties of its solutions is provided. This includes the case of elliptic equations with timelike independent variables: Their solutions acquire dynamical properties (more exactly, behave as solutions of parabolic equations). The following approach to classify differential-difference equations into types, based on the property of differential-difference operators to be Fourier multipliers is applied in the following manner: An operator is treated to be elliptic if the real part of its symbol is positive, while the parabolic and hyperbolic types are defined correspondingly. It is shown that the proposed approach (being a natural extension of the classical notion of the ellipticity) is reasonable. On the other hand, fundamental novelties (compared with the classical theory of partial differential equations) occur as well. We provide conditions which guarantee the following results. For the half-space (half-plane) Dirichlet problem for elliptic equations, integral representations (of the Poisson type) of solutions are constructed, which are infinitely differentiable outside the boundary hyperplane (plane), and the asymptotic closeness of solutions (as the timelike independent variable unboundedly increases) takes place. For the Cauchy problem for parabolic equations, the same is valid, but we deal with the classical time instead of the timelike independent variable. For hyperbolic equations, multiparameter families of infinitely smooth global solutions are constructed. The said (sufficient) conditions restrict the sign of the real part of the symbol for the differential-difference operator with respect to spatial or spacelike independent variables. In a number of special cases, they might be weakened such that symbols with sign-changing real parts are admitted. The objective of the study is to observe the current stage of the classification issues for partial differential-difference equations in terms of the aforementioned approach: The ellipticity of a Fourier multiplier is defined by means of the sign (of there real part) of its symbol. Since both differential and translation operators are Fourier multipliers, methods of Fourier analysis are applicable in this study: We apply the Fourier transformation to the original partial differential-difference equation, solve the obtained ordinary differential equation, and apply the inverse Fourier transformation to the obtained solution. The main contribution obtained within this study is an efficient (workable) type concept for the fundamentally new extension of the class of partial differential equations, which is the class of partial differential-difference equations.