Regularization for ill-posed Cauchy problems associated with generators of analytic semigroups
Consider an initial value problem like the backward heat equation \[ u'(t)=Au(t),\;0<t<T,\quad u(0)=f.\tag{1} \] If \(-A\) generates an analytic semigroup \(S=\{S(t):t>0\}\), then the unique solution of (1) is, for suitable choices of \(f\), \(u(t)=U(t)f\), where \(U(t)\) is the inverse of \(S(t)\), and for \(t>0\). \(U(t)\) is an unbounded operator (unless \(A\) is bounded, which one assumes not to be the case). Larry Payne and many others showed how to make these ill-posed problems well-posed, by demanding that solutions satisfy other sorts of constraints (often involving uniform boundedness). The method of quasi-reversibility [\textit{R. Lattes} and \textit{J.-L. Lions}, Méthode de quasi-réversibilité et applications. Travaux et Recherches Mathématiques, 15. Paris: Dunod (1969; Zbl 0159.20803)], gives a way to regularize \(A\); it replaces \(A\) by \(A-aB\), where \(B\) is the \(b\)-th (fractional) power of \(A\); here \(a>0\) and \(b>1\). \textit{J. A. Goldstein} (1969) showed that \(-B\) generates an analytic semigroup if the sector in which \(S\) is analytic has angle \(>\pi/4\). In this case one may take \(b=2\), but in general a smaller \(b\) is needed. The authors show that for a suitable class of initial data \(f\) for which the solution of (1) exists, one can approximate \(u(t)\) by a sequence \(v(t)\) corresponding to replacing \(A\) by \(A-aB\) where \(a\) tends to zero through an appropriate sequence. The regularizing operators are constructed via the Lattes-Lions quasi-reversibility method. The results are interesting.
scientific article; zbMATH DE number 2093537
scientific article