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Area of Science:

  • Partial Differential Equations
  • Mathematical Analysis
  • Harmonic Analysis

Background:

  • Higher-order hyperbolic pseudo-differential equations present challenges in analysis.
  • Understanding well-posedness is crucial for solving related Cauchy problems.

Purpose of the Study:

  • To establish sufficient conditions for the well-posedness of higher-order hyperbolic pseudo-differential equations.
  • To analyze equations with variable multiplicities and time-dependent principal parts.

Main Methods:

  • Transformation into a first-order system.
  • Reduction to upper-triangular form.
  • Application of Fourier integral operator methods for non-diagonalizable systems.

Main Results:

  • Identification of specific Levi conditions on roots and lower-order terms.
  • Demonstration of well-posedness for the Cauchy problem under these conditions.
  • Analysis in arbitrary space dimensions with time-dependent principal parts.

Conclusions:

  • The derived Levi conditions are sufficient for well-posedness.
  • The methods extend existing literature on hyperbolic equations.
  • Provides a framework for analyzing complex hyperbolic pseudo-differential equations.