Time Reversal in Abstract Cauchy Problems

Abstract

Using the fact that under homogeneous endpoint conditions on a finite interval the eigenvalues of the operator ${{d^2 } / {dx^2 }}$ tend to $ - \infty $, a standard construction shows that initial boundary value problems for the backward heat equation $u_x = - u_{xx} $ are not well-posed in the sense that the solutions do not depend continuously on the data. Corresponding problems for the forward heat equation are well-posed. Abstract considerations show that the pathology of heat conduction problems is essentially due to the unboundedness of ${{d^2 } / {dx^2 }}$ rather than to the distribution of its eigenvalues. Indeed, if A is a closed, unbounded operator in a Banach space X, and if the initial value problem $u'(t) = Au(t)$, $t > 0$, $u(0) = f$, is well-posed in X, then the backward initial value problem $u'(t) = - Au(t)$, $t > 0$, $u(0) = f$, is not well-posed, even with the data f restricted to the domain $D(A)$ of A.