Higher-order heat and Laplace-type equations with real time variable and complex spatial variable

Abstract

It is known that, if the time variable in the heat equation is complex and belongs to a sector in ℂ, then the theory of analytic semigroups becomes a powerful tool of study. The same is true for the Laplace equation on an infinite strip in the plane, regarded as an initial value boundary value problem. 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. In a recent paper (C.G. Gal, S.G. Gal, and J.A. Goldstein, Evolution equations with real time variable and complex spatial variables, Complex Var. Elliptic Eqns. 53 (2008), pp. 753–774), a complementary approach was made: the study of the complex versions of the classical heat and Laplace equations, obtained by ‘complexifying’ the spatial variable only (and keeping the time variable real). The goal of this article is to extend that study to the higher-order heat and Laplace-type equations. This ‘complexification’ is based on integral representations of the solutions in the case of a real spatial variable, by complexifying the spatial variable in the corresponding semigroups of operators. It is of interest to note that these solutions preserve some geometric properties of the boundary function, like the univalence, starlikeness, convexity and spirallikeness.