Evolution equations with real time variable and complex spatial variables

Abstract

It is known that if the time variable in the heat and wave equations is complex and belongs to a sector in ℂ, then the theory of analytic semigroups becomes a powerful tool of study. Also, it is known that if both variables, time and spatial, are complex, then e.g. the Cauchy problem for the heat equation admits as solution, only a formal power series which, in general, converges nowhere. The purpose of this article is, in a sense, complementary: to study the complex versions of the classical heat and Laplace equations, obtained by ‘complexifying’ now the spatial variable only (and keeping the time variable real). This ‘complexification’ is studied by two different methods, which produce different equations: first, one complexifies the spatial variable in the corresponding semigroups of operators and secondly, one complexifies the spatial variable in the corresponding evolution equation and then one searches for analytic and non-analytic solutions. It is of interest to note that in the case of the first method, besides the fact that the solutions can be studied by using the theory of semigroups of linear operators, also these solutions preserve some geometric properties of the boundary function, like the univalence, starlikeness, convexity and spirallikeness. Also, by both methods new kinds of evolution equations (or systems of equations) in two-dimensional spatial variables are generated and their solutions are constructed.