Quantitative unique continuation for a parabolic equation

Abstract

We address the quantitative uniqueness properties of the solutions of the parabolic equationwhere v and w are bounded. We prove that for solutions u, the order of vanishing is bounded by C( v 2/3 L ∞ + w 2 L ∞ ) matching the upper bound previously established in the elliptic case.

On quantitative uniqueness for parabolic equations

“…The lemma is proven in [CK,Ku4]; we provide a short argument for the sake of completeness. In the sequel, we reserve ǫ for the time provided in Lemma 3.1.…”

“…It is wellknown that sp(H) = {m/2 : m ∈ N 0 }, (cf. [CK,p.664]). Thus we obtain Q(τ ) → m/2 as τ → ∞ for some m M a 0 + M b 1 .…”

“…The first is to find the point in space where the frequency function is the smallest and translate the equation so it starts at that point (cf. Lemma 3.1 below); this idea has been introduced in [Ku4,CK]. The second device is to use the embedding theorems with Laplacian and use (3.24)-(3.25) below to bound the parts containing v and w. The third device is to use the finiteness of the integral in (3.47) below, which then allows us to show the convergence of the quantity under the integral.…”

A local asymptotic expansion for a solution of the Stokes system

“…[AE, AMRV, CRV, DF, EFV, EV, GL, JK, KT, SS1, SS2] for instance); for more complete reviews, see [K1, K2, V]. Unique continuation questions for the Stokes and Navier-Stokes systems were addressed in [CK,FL1,FL2,Ku]. The paper is organized as follows.…”

A unique continuation property for a class of parabolic differential inequalities in a bounded domain

“…More recently, G. Camliyurt and I. Kukavica in [3] pursued the unique continuation by checking finite orders of vanishing for forward parabolic PDEs with 1st derivative term, whose coefficients are variable but bounded. Frequency functions and a technique of changing variables are invoked in their work.…”