ON THE FUNCTIONS RELATED WITH THE ANALYTIC SOLUTIONS OF THE CAUCHY PROBLEM FOR WAVE AND HEAT EQUATIONS

Abstract

We investigate properties of entire solutions of the Cauchy problem for one-dimensional homogeneous hyperbolic equation.  Considering analytic continuation of the solutions given by the D'Alambert formula we have found some conditions providing $L$-index boundedness in the direction for some functions related with the solutions.  In particular, for homogomogeneous  wave equation $c^2\frac{\partial^2 }{\partial x^2}u(x,t)=\frac{\partial^2 }{\partial t^2}u(x,t)$  with initial conditions $u(x,0)=\varphi (x),$ $u_t(x,0)=\psi (x)$ its solution has the form $u(x,t)=\frac{\varphi(x+ct)+\varphi(x-ct)}{2}+\frac{1}{2c}\int\limits^{x+ct}_{x-ct}{\psi(\alpha)d \alpha}.$   We study the functions $\mathfrak{H}(x,t)=\frac{\varphi(x+ct)+\varphi(x-ct)}{2}+ \frac{\mathfrak{E}}{2c}\int\limits^{x+ct}_{x-ct}{\psi(\alpha)d \alpha},$ where $x,$ $t,$ $c\in\mathbb{C},$ $\mathfrak{E}$ is a positive constant which is determined with some conditions by  the functions $\varphi$ and $\psi$. Our main result gives sufficient conditions of boundedness of $\mathfrak{L}$-index in a direction $\mathbf{b}$ for the functions $\mathfrak{H}$. Its proof uses known sufficent conditions for the sum of entire functions.   At the end, we pose open problems concerning conditions of the directional $L$-index boundedness for analytic solutions of the Cauchy problem of the heat equation. The conditions will allow a qualitative description of  local and asymptotic behavior of the parabolic equation analytic solutions presenting the temperature distribution in the process of plasma electrolytic oxidation.