Degenerate high-order differential equations of a special kind in Banach spaces and their applications

Using generalized function methods in Banach spaces, we study the solvability of Cauchy problems for third or fourth order differential equations with a Fredholm operator at the highest derivative in the class of distributions with support bounded on the left. In terms of the generalized Jordan structure of the degenerate principal part of these equations we construct fundamental operator functions that correspond to the differential operators and use them to recover generalized solutions, prove uniqueness pf the latter, and study relations with the classical solutions. We apply the results to two initial-boundary value problems for nonclassical equations of mathematical physics. We study the Cauchy–Dirichlet problem for the generalized electric potential equation and the thermoelastic plate equation.