Solution of the Initial Value Problem for a Linear Evolution Equation in a Clifford Type Algebra

Initial value problems of type$$\partial_t{u} = L(t, x, u, \partial_{{x}_i}u), $$∂tu=L(t,x,u,∂xiu),$$u(0, x) = \varphi(x), $$u(0,x)=φ(x),where t is the time, L is a linear first order operator in a Clifford Analysis and φ is a generalized metamonogenic function, 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 formulates sufficient conditions on the coefficients of operator L under which L is associated to differential equations with anti-monogenic right-hand sides. We shall exhibit an interior estimate for the generalized metamonogenic functions using the Cauchy integral formula and then give out conditions under which this initial value problem (0.1), (0.2) is uniquely solvable in the context of Clifford type algebras.