Stabilized approximate solutions of the inverse time problem for a parabolic evolution equation (Q1094654)

Let H be a real Hilbert space, A a self-adjoint positive linear operator, and B a self-adjoint bounded linear operator on H. Consider the differential equation in H: \[ (1)\quad u'(t)=-Au(t)+Bu(t). \] The following problem: given a \(\chi\in H\), find a solution u(t) of (1) such that \(u(1)=\chi\), is not well-posed. A one-parameter family \(v_{\beta}(t,\chi)\) of solutions of (1), stable with respect to variations in \(\chi\) and such that \(v_{\beta}(0,\chi)\to u(0)\) and \(v_{\beta}(1,\chi)\to \chi\) for \(\beta\to 0\), is called a quasi- solution of the problem. In this paper the author studies the quasi- solution obtained in the following way: Let \(S(t)\), \(t>0\), be the \(C_ 0\) semigroup generated by \(-A+B\), and let \(S(1)=K\). Denote \(v_{\beta}(t,\chi) = S(t)(\beta I+K)^{-1}\chi\), where \(\chi\in H\) and \(\beta >0\). Then \(v_{\beta}(t,\chi)\) is a quasi-solution of (1) having the following property: If \(u(t)\) is a solution of (1) satisfying \(| u(1)-\chi | <\epsilon\) and \(| u(0)| <E\), then, for \(\beta = \epsilon /E\), we have \(| v_{\beta}(t,\chi)-u(t)| \leq 2\epsilon^ tE^{1-t}\alpha^ t\), where \(\alpha = \| K\| /(\beta +\| K\|)\).