Weak solutions for a class of quadratic operator-differential equations

Abstract

"This monograph presents results of existence and uniqueness of generalized solutions for two classes of quadratic operator-differential equations with constant coefficients: (1) Au(t) + Bu'(t) - Du"(t) = 0 and (2) Au(t) + iBu'(t) + Da"(t) = 0, where A, B and D are self-adjoint operators which satisfy certain conditions under which the equation (1) is called elliptic-hyperbolic and the equation (2) is called hyperbolic. The main result is the existence and uniqueness of weak solutions, on the positive real axis or on the negative real axis, for the class of operator differential equations called elliptic-hyperbolic. These solutions, on the positive real axis, decay exponentially to zero in the infinity. A similar result is obtained on the negative real axis..."