On a System of Linear Singular Partial Differential Equations with Weight Functions

Let X be a Banach space, Omega an open bounded subset of X, and Y a complex Banach space. We consider a Volevic system of singular linear partial differential equations of the form t partial derivative u(i)/partial derivative t = Sigma(N)(j=1) a(ij)(t, x)u(j)(t, x) + Sigma((j),(k)is an element of N(i)) b(jk)(t, x)((mu(0)(t)D)(k)u(j)(t, x) . x(k)((k)))(j, k) + g(i)(t, x), (1) 1 <= i <= N, in the unknown function u = (u(1), u(2),.., u(N)) is an element of Y-N of t >= 0 and x is an element of Omega, where a(ij), b(jk) is an element of C, x(k) = (x,.., x) (x is k times) D denotes the Frechet differentiation with respect to x, and N(i) = {(j, k) : j and k are integers; 1 <= j <= N, 0 < k <= n(i, j)}, (2) n(i, j) = n(i) - n(j) + 1, where n(i), i = 1, 2,..., N, are nonnegative integers. The map mu(0) belongs to C-0([0, T], C). We express growth estimates in terms of weight functions and we establish an existence and uniqueness theorem for our system in the class of ultradifferentiable maps with respect to the space variable x.