Weak regularization for a class of ill-posed Cauchy problems (Q2507634)

The authors consider the ill-posed Cauchy problem \[ u'(t)=Au(t), \quad 0<t<T, \quad u(0)=x, \] where \(A\) is a densely defined linear operator in a Banach space, the spectrum of \(A\) is contained in a sector of right-hand complex plane, and its resolvent is polynomially bounded. They present the weak regularization for such ill-posed Cauchy problem by using the quasi-reversibility method and analytic regularized semigroups. The results are illustrated by an example.