The Cauchy Problem

abstract Cauchy problem adjoint Akad analytic apply approximation arbitrary argument assume assumptions Au(t B₁ Banach space belongs bounded operator Cauchy problem coefficients conditions of type continuous function continuously differentiable convergence Corollary deduce definition denote densely defined operator depend differential operators Dirichlet boundary condition dissipative operators domain duality map E-valued ends the proof equation u'(t estimate Example exists a constant extension fact following result formula hence Hilbert space holds hypotheses implies inequality integral Lemma linear LP norm m-dissipative Math Nauk nonnegative norm obtain obvious partial differential perturbation proof of Theorem propagator properly posed properties prove resp right-hand side Russian satisfies Section self-adjoint semigroups sequence solution strongly continuous subspace symmetric theory u₁ uniformly on compact value problem zero