arXiv subjects

Ti-Jun Xiao

Publications and source records attributed to Ti-Jun Xiao.

4 recordsLinked to original sources

arXiv2026

Dominance of slow solutions for second order abstract evolution equations with time-varying damping

Of concern is a class of non-autonomous evolution equations of second order in Hilbert spaces, with a nonnegative self-adjoint operator $A$, time-varying damping and nonlinear source term. We give an upper decay rate of the energy, valid for all solutions and solely based on the damping coefficient and the geometrical index of the source term. Furthermore, we prove under suitable conditions that for all initial data, except for those in the kernel of $A$, the solutions decay (in the energy norm) at most as fast as this decay rate. The result not only shows the optimality of the decay rate, but also reveals an unusual phenomenon: ``slow solutions", i.e. those that decay at {\it exactly} this rate, are dominant in amount. Moreover, specialized to the case when the nonlinear source is absent, our result improves relevant existing ones to a large extent.

math.AP↗

arXiv2026

Regularity of global attractors for beam equations with fractional damping and memory

This paper investigates the long-time behavior of a semilinear beam equation in a domain $\Omega \subset R^{n}$, with memory and fractional damping of the form $(-\Delta)^{\alpha}u_{t}$ ($\alpha \in [0,2]$ the dissipation index). Two critical growth indices of the nonlinear term are determined for smooth and $C^2$ boundaries respectively, concerning the existence of the associated semigroup. We prove the existence of global attractor for the semigroup by showing that it possesses a bounded absorbing set and asymptotic compactness. Furthermore, we find out a new way to obtain, for all $\alpha$, higher regularities than anticipated for the attractors, and the regularity result indicates an interesting phenomenon that even much weaker damping can produce regularity that is infinitely close to that in the case of strong damping ($\alpha =2$). As a consequence, our regularity result deepens and extends the existing related ones for the case when the memory is absent.

math.AP↗

arXiv2024

Sharp $L^1$-convergence rates to the Barenblatt solutions for the compressible Euler equations with time-varying damping

We study the asymptotic behavior of compressible isentropic flow when the initial mass is finite and the friction varies with time, which is modeled by the compressible Euler equation with time-dependent damping. In this paper, we obtain the best $L^1$-convergence rates to date, for any $\gamma\in(1,+\infty)$ and $\nu\in[0,1)$. Here, $\gamma$ is the adiabatic gas exponent, and $\nu$ is the physical parameter in the damping term. The key to the analysis lies in a new perspective on the relationship between the density function and the Barenblatt solution of the porous medium equation, and finding the relevant lower bound for the case of $\gamma<2$ is a tricky problem. Specialized to $\nu=0$, these convergence rates also show an essential improvement over the original rates. Moreover, for all $\gamma\in(1,+\infty)$, the results in this work are the first to present a unified form of $L^1$-convergence rates. Indeed, even for $\nu=0$, as noted in 2011, ``the current rate is difficult to improve with the current method". Our results are therefore an encouraging advancement.

math.AP↗

arXiv2024

Sharper L^1-convergence rates of weak entropy solutions to damped compressible Euler equations

We consider the asymptotic behavior of compressible isentropic flow when the initial mass is finite, which is modeled by the compressible Euler equation with frictional damping. It is shown in \cite{HUA} (resp.\cite{GEN}) that any $L^{\infty}$ weak entropy solution of damped compressible Euler equation converges to the Barenblatt solution with finite mass in $L^1$ norm, with convergence rates depending on the adiabatic gas exponent $\gamma$ in the case of $1<\gamma<3$ (resp.$\gamma\ge2$). Whether or not these convergence rates can be improved remains an interesting and challenging open question. In this paper, we obtain a better $L^1$ convergence rate than that in \cite{GEN}, for any $\gamma\ge2$, through a new perspective on the relationship between the density function and the Barenblatt solution of the porous medium equation. Furthermore, making intensive analysis of some relevant convex functions, we are able to obtain the same form of $L^1$ convergence rate for $1<\gamma<9/7$, which is better than that in \cite{HUA} as well.

math.AP↗