Initial value problems in Clifford-type analysis

We consider an initial value problem of type $$\frac{\mathrm{\partial }u}{\mathrm{\partial }t}=\mathcal{F}(t,x,u,{\mathrm{\partial }}_{j}u),u(0,x)=\varphi (x),$$ where $t$ is the time, $x\in {\mathbb{R}}^n$ and $u_0$ is a Clifford type algebra-valued function satisfying ${\bf D}u=\displaystyle\sum_{j=0}^{n}\lambda_j(x)e_j\partial_ju = 0$,$\lambda_j(x)\in $\mathbb{R}$ $forall$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 $\mathcal{F}(u)={\displaystyle \sum _{i=0}^n{A}^{(i)}(x){\displaystyle {\mathrm{\partial }}_{i}u}}$ 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.