Backward uniqueness - Knowlpedia

An evolution problem has backward uniqueness in a specified class if any two solutions that agree at a time $T$ agree throughout their common earlier time interval. For a linear evolution family $S(T,s)$ , this is injectivity of $S(T,s)$ .

Heat equation example

For the heat semigroup on $L^2(\mathbb R^n)$ , the Fourier multiplier is $e^{-4\pi^2\nu(T-s)|\xi|^2}>0$ . Thus $S(T,s)v=0$ implies $v=0$ . Inverting this multiplier is unbounded on $L^2$ : uniqueness of past values does not give existence or continuous dependence for arbitrary terminal data. Backward uniqueness for equations with variable coefficients requires additional hypotheses.