Abstract
Initial value problems (IVPs) of type $$\frac{\partial u}{\partial t} = L ( t, x, u, \frac{\partial u}{\partial x_j} ), \quad u(0, x) = \varphi (x)$$ can be solved by applying the method of associated spaces which is constructed by W.Tutschke (Teubner Leipzig and Springer-Verlag 1989). The present paper considers above IVPs in the space of Helmholtz-type generalized regular functions in the sense of quaternionic analysis. Using the Poisson integral formula, we shall prove an interior estimate for Helmholtz-type generalized regular functions and then give out conditions under which these IVPs are uniquely solvable.
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