Hardy Function and Unique Continuation for Evolution Equations

In this paper, unique continuation problems are considered for the first order evolution equation: ut(x, t) = P(t, D) u(x, t), x ∈ Rn, t ∈ R, (∗) where P(t, D) represents a mth order differential operator with time dependent coefficients. In one space dimension case (n = 1), a necessary and sufficient condition is given for any nonzero solution of (∗) in the class of C(R; L2(R)) to have a support on a horizontal half line in the x−t space at two different times. With some assumptions on the coefficients of P(t, D), it is shown that any solution u(x, t) ∈ C(R; L2(R)) is uniquely determined by its part on any two horizontal half lines or any open subset in the x−t space. In higher space dimension case (n > 1), a necessary condition is given for any solution u ∈ C(R; L2(Rn)) of (∗) to be supported on a half hyperplane in the space Rn at two different times. Some higher order evolution equations are also considered for their unique continuation properties.