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.

Keywords:

• Carleman estimate
• higher order
• parabolic equations
• Schrödinger equations
• unique continuation

2010 Mathematics Subject Classification:

• 35G05
• 35K25
• 35A02
• 35B60

Acknowledgements

The author was visiting The University of Chicago (from Huazhong University of Science and Technology, supported by the China Scholarship Council), while this research was carried out, and he thanks Carlos E. Kenig for fruitful discussion on this work. The author also thanks Quan Zheng for some helpful discussion on related topics, and the anonymous referee for the advice on the exposition of this manuscript.