The study of large-time behavior of solutions to partial differential equations is a fundamental pursuit in mathematical analysis, with profound implications for physics and engineering. It addresses a core question: regardless of the initial data, will the solutions eventually settle into a simple, predictable pattern? Answering this question is crucial for verifying the long-term validity of mathematical models and predicting final, stable states. Asymptotic states—such as shock waves, rarefaction waves, and contact waves—are universal patterns that serve as fundamental building blocks.
In a study led by Mr. Mutong He, Professor Feimin Huang (Academy of Mathematics and Systems Science, Chinese Academy of Sciences), and Professor Tian-Yi Wang (Wuhan University of Technology), the asymptotic behavior of global solutions to the damped wave equation with partially linearly degenerate flux is studied. The authors show that the global solution converges to a combination of a rarefaction wave and a viscous contact wave as time tends to infinity.
While the asymptotic behavior for strictly convex fluxes is well understood, the case where the flux includes the partially linearly degenerate introduces significant challenges due to the loss of uniform convexity and the failure of standard methods. To overcome these difficulties, the key innovations are as follows:
This work presents the first asymptotic stability result for multi-wave patterns in damped wave equations with the partially linearly degenerate flux. The analysis requires neither the initial perturbation nor the wave strength to be small. The methods developed here are expected to be applicable to other problems involving nonconvex fluxes.
See the article:
Asymptotic Stability of Global Solutions for a Class of Semilinear Wave Equation https://doi.org/10.1007/s10473-025-0617-5
Acta Mathematica Scientia
Asymptotic stability of global solutions for a class of semilinear wave equation
29-Oct-2025