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:
- The propagation speed of the viscous contact wave cannot be normalized, unlike in viscous conservation laws. Therefore, the viscous coefficient must be designed in relation to the propagation speed. The sub-characteristic condition ensures that this coefficient is positive, which is essential for constructing the correct asymptotic state.
- The equation could be reformulated to the Jin–Xin relaxation system, leading to uniform boundedness of solutions. This avoids the need for smallness conditions on either the initial perturbation or the wave strength.
- A domain-partitioned weighted energy estimate is developed to handle the loss of convexity. This method divides the integration domain according to the flux and performs estimates in these regions separately.
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
