Initial value problems in Clifford-type analysis

Abstract

We consider an initial value problem of type $$\frac{\partial u}{\partial t}=\mathcal{F}(t,x,u,{\partial }_{j}u),u(0,x)=\varphi (x),$$ where $t$ is the time, $x\in {\mathbb{R}}^n$ and $u_0$ is a Clifford type algebra-valued function satisfying $\mathbf{D}u=\sum _{j=0}^n{\lambda }_j(x)e_j{\partial }_{j}u=0$, ${\lambda }_j(x)\in \mathbb{R}$ for all $j$. We will solve this problem using the technique of associated spaces. In order to do that, we give sufficient conditions on the coefficients of the operators $\mathcal{F}$ and $\mathbf{D}$, where $\mathcal{F}(u)=\sum _{i=0}^n{A}^{(i)}(x){\partial }_{i}u$ for $A^{(i)}(x)\in \mathbb{R}$ or $A^{(i)}(x)$ belonging to a Clifford-type algebra, such that these operators are an associated pair.