Cauchy problem for quasihyperbolic factorized differential equations with variable domains of discontinuous operators

Abstract

Quasuhyperbolic factorized differential-operator equations with variable domains of smooth operator coefficients were considered in [1]. In the case of discontinuous operator coefficients, only hyperbolic second-order differential-operator equations were investigated [2,3]. In the present paper, we prove the strong well-posedness and the smoothness of strong solutions of quasihyperbolic factorized differential-operator equations with variable domains of discontinuous operator coefficients; i.e., one of the main results for hyperbolic second-order differential-operator equations with discontinuous operators in [4] is generalized to the case of quasihyperbolic even-order differentialoperator equations.

Boundary value problems for complete partial differential equations of variable order

“…Boundary value problems for parabolic equations in variable-order partial derivatives (in space variables) were earlier considered for higher derivatives of the first order [1][2][3] and odd order [4] in time, for hyperbolic equations with higher derivatives of the second order [3] and even order [5] in time, for partially pseudodifferential parabolic equations with higher derivatives of the first order [6] and higher order [7] in time, and for partially pseudodifferential hyperbolic equations with even-order derivatives in time [8]. In the present paper, we prove the strong well-posedness of new boundary value problems for partially pseudodifferential complete nonclassical equations of variable order with odd-order higher derivatives in time.…”

“…In the present paper, we prove the strong well-posedness of new boundary value problems for partially pseudodifferential complete nonclassical equations of variable order with odd-order higher derivatives in time. The variable differentiation order depended on the time in [1][2][3][4][5] and on space points at which the differentiation is performed in [6,7].…”