Journal of Hyperbolic Differential Equations

We address here a viscoelastic Timoshenko model on the (one-dimensional) real line with memory damping coupled on a shear force. Our main results concern a complete decay structure of the system under the so-called equal wave speeds assumption, as well as without this condition. This is the first result of this type for partially dissipative beam systems with memory-type damping on the shear force. Our method is based on expanded structural conditions such as the so-called SK condition. In addition, we give a characterization of the dissipative structure of the system by using a spectral analysis method, which confirms that our decay structure is optimal.

We consider a singular limit problem from the damped wave equation with a power type nonlinearity (NLDW) to the corresponding heat equation (NLH). We call our singular limit problem non-delay limit. We show that the solution of NLDW goes to the one of NLH in $H^1$ topology under the both $H^1$ regularity solutions. We also obtain the positive convergence rate in the weaker topology $L^2$. Moreover, with restriction of the range of power, if the solution to NLH is global and decays to zero, then we get the global-in-time uniform convergence of the non-delay limit.

In this paper, we consider the Cauchy problem for higher-order weakly hyperbolic equations assuming that the principal symbol depends only on one space variable and the characteristic roots ${\tau }_j$ verify an inequality like

This paper is devoted to a hyperbolic system of nonlinear conservation laws, that is, the pressure system independent of density and energy from the Liou–Steffen flux-splitting scheme on the compressible Euler equations. First, the one-dimensional Riemann problem is solved with eight kinds of structures. Second, the two-dimensional Riemann problem is discussed; the solution reveals a variety of geometric structures; by the generalized characteristic analysis method and studying the pointwise interactions of waves, we construct 29 kinds of structures of solution consisting of shocks, rarefaction waves and contact discontinuities; the theoretical analysis is confirmed by numerical simulations.

In this paper, we discuss the existence, uniqueness and asymptotic stability of global piecewise $C^1$ solution to the mixed initial-boundary value problem for 1-D quasilinear hyperbolic systems on a tree-like network. Under the assumption of boundary dissipation, when the given boundary and interface functions possess suitably small $C^1$ norm, we obtain the existence and uniqueness of global piecewise $C^1$ solution. Moreover, when they further possess a polynomial or exponential decaying property with respect to $t$, then the corresponding global piecewise $C^1$ solution possesses the same or similar decaying property. These results will be used to show the asymptotic stability of the exact boundary controllability of nodal profile on a tree-like network.