Generalized Analyticity and Some Related Properties of Solutions of Elliptic and Parabolic Equations

Doubling properties of caloric functions

“…On the other hand, the recent applications to stability issues in Inverse Problems of quantitative estimates of unique continuation ( [2], [3]) show it is useful to know that "harmonic"functions are as small as they can be at intermediate scales when they are known to be small at smaller scales. The first quantitative result of strong unique continuation for parabolic equations, a two-sphere one-cylinder inequality [13,Theorem 9 ′ ] (see the body of the paper for the relevant definitions), was derived in 1974 from the corresponding first qualitative results of strong unique continuation for second order parabolic equations in the literature [13]. In [13], E.M. Landis and O.A.…”

The Rate of Spatial Decay of Nonnegative Solutions of Nonlinear Parabolic Equations and Inequalities

“…Let L be the uniformly parabolic linear partial differential operator defined Landis and Oleinik [6] conjectured that bounded solutions of the equation…”

“…The proof of this is very similar to the proofs of [7, Theorems 1 and 2], and hence we merely give an outline here. Let M(r) = max{v(x, T) : \\x\\ = r} and proceed as in [7] to show that inequalities (5), (6), and (7) are incompatible with the existence of jc0 £ R" for which v(x0, T) > 0. (Our a corresponds to a /4 in [7].)…”

“…Suppose v £ C*(D) is a nonnegative function satisfying (6) [ r(x,t;t,z)v(t,T)dZ<v(x,t), 0<r<t<T, and (7) v(x, T) < Ce~iM~ fior some C> a/T and all x£R" .…”

“…We show that nonnegative solutions of the differential inequality Lu < c(u + |V«|) on R" x (0, T) for which u(x, T) = 0(exp(-<5|x|2)) must be identically zero if the constant ô is sufficiently large. An analogous result is given for nonlinear systems.Let L be the uniformly parabolic linear partial differential operator defined Landis and Oleinik [6] conjectured that bounded solutions of the equationwhich decrease on a characteristic at the rate exp(-|x|~"_£), e > 0, must be identically zero. Since then a handful of authors has successfully solved this and related problems.…”