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

Abstract

Let X be a Banach space, Ω an open bounded subset of X, and Y a complex Banach space. We consider a Voleviˇc system of singular linear partial differential equations of the form t ∂ui ∂t = X N j=1 aij (t, x)uj (t, x) + X (j,k)∈N(i) bjk(t, x)((µ0(t)D) kuj (t, x) · x (k) k )(j,k) + gi(t, x), (1) 1 ≤ i ≤ N, in the unknown function u = (u1, u2, ..., uN ) ∈ Y N of t ≥ 0 and x ∈ Ω, where aij , bjk ∈ C, xk = (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 µ0 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.