Abstract
We prove that a matrix-differential operator A in a Banach space is a Fredholm operator with zero index and construct the corresponding subspaces and projections. We find the spectrum of A. Based on properties of the operator A, we study solutions to the initial-boundary value problem for a second order partial differential equation with a small parameter e at the higher order derivatives and some parameter c at lower order terms. We clarify how the parameter c affects the behavior of the solution as e → 0.
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