Unique continuation through hyperplane for higher order parabolic and Schrödinger equations

Abstract

Consider the higher order parabolic operator ${\partial }_t+(-{\Delta }_x{)}^m$ and the higher order Schrödinger operator $i^{-1}{\partial }_t+(-{\Delta }_x{)}^m$ in $X=\{(t,x)\in {\mathbb{R}}^{1+n};|t|<A,|x_n|<B\}$, where $m$ and $n$ are any positive integers. Under certain lower order and regularity assumptions, we prove that if solutions to the linear problems vanish when $x_n>0$, then the solutions vanish in $X$. Such results are global if $n>1$, and we also prove some relevant local results.

Mainly rewrite the introduction. Also correct some inaccuracy

References in corpus (1)

• Spectral inequality and resolvent estimate for the bi-Laplace operator

Cited by in corpus (1)

• Unique continuation properties for one dimensional higher order Schrödinger equations