Approximate solutions to the abstract Cauchy problem (Q2708551)

The notion of approximate solution to the abstract Cauchy problem NEWLINE\[NEWLINE u'(t)=Au(t), \quad u(0)=x, \quad 0\leq t< T\leq \infty NEWLINE\]NEWLINE is introduced. The operator \(A\) is linear and closed on a Banach space \(X\) and \(x\in X\) is given. The equivalence of this notion with the regularized solutions introduced by I. Cioranescu and G. Lumer is shown. As an example, it is studied the backwards heat equation in \(\Omega\subset\mathbb{R}^d\). This problem is well-posed in the approximate sense if and only if \(d=1\).NEWLINENEWLINEFor the entire collection see [Zbl 0957.00037].