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Quantitative unique continuation for the heat equations with inverse square potential.

Guojie Zheng1, Keqiang Li1, Yuanyuan Zhang2

  • 11College of Mathematics and Information Science, Henan Normal University, Xinxiang, P.R. China.

Journal of Inequalities and Applications
|March 7, 2019
PubMed
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This study explores unique continuation for multi-dimensional heat equations. Solutions can be uniquely determined by their values on a subset of the domain at any positive time.

Keywords:
Frequency functionHeat equationsSingular potentialUnique continuation

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Area of Science:

  • Partial Differential Equations
  • Mathematical Physics
  • Harmonic Analysis

Background:

  • Unique continuation problems are crucial for solving inverse problems in various scientific fields.
  • Heat equations with inverse square potentials present unique mathematical challenges due to the singularity of the potential.

Purpose of the Study:

  • To investigate the unique continuation properties of multi-dimensional heat equations with an inverse square potential.
  • To establish observation estimates for solutions within a bounded convex domain.

Main Methods:

  • Utilizing techniques from the theory of partial differential equations.
  • Developing specific analytical tools to handle the inverse square potential.
  • Establishing rigorous mathematical proofs for the established estimates.

Main Results:

  • Demonstrated unique continuation properties for the considered heat equations.
  • Derived novel observation estimates for the solutions.
  • Showcased that solution values are uniquely determined by partial data.

Conclusions:

  • The findings contribute to a deeper understanding of unique continuation for degenerate parabolic equations.
  • The results have potential applications in inverse problems and data assimilation.
  • This research advances the mathematical framework for analyzing heat equations with singular potentials.