Solution of Initial Value Problems of Cauchy-Kovalevsky Type in the Space of Generalized Monogenic Functions

This paper deals with the initial value problem of the type (0.1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial _{t} u(t,x) = {\mathcal{L}}u \left( {t,x} \right), \quad u(0,x) = u_0(x)$$\end{document} where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t \in \mathbb{R}^+_0$$\end{document} is the time, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x \in \mathbb{R}^{n+1}, \, u_{0}(x)$$\end{document} is a generalized monogenic function and the operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{L}$$\end{document}, acting on a Clifford-algebra-valued function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u\left({t,x} \right) = \sum\limits_{B} u _{B} \left( {t,x} \right)e_{B}$$\end{document} with real-valued components uB(t, x), is defined by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{L}}u(t, x) := \sum\limits_{A,B,i} {c_{B,i}^{(A)} \left(t,x \right) \partial _{x_{i}} u_{B} \,\left( t,x \right)e_A }+ \sum\limits_{A,B} {d_{B}^{(A)}} \left( t,x \right)\,u_{B}\left( {t,x} \right)e_{A} + \sum\limits_{A} g_{A} \left( t,x \right)e_{A} $$\end{document} We formulate sufficient conditions on the coefficients of the operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{L}}$$\end{document} under which \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{L}}$$\end{document} transforms generalized monogenic functions again into generalized monogenic functions. For such an operator the initial value problem (0.1) is solvable for an arbitrary generalized monogenic initial function u0 and the solution is also generalized monogenic for each t.