Journal of Hyperbolic Differential Equations

The aim of this paper is to prove that, for specific initial data $(u_0,u_1)$ and with homogeneous Neumann boundary conditions, the solution of the IBVP for a hyperbolic variation of Allen–Cahn equation on the interval $[a,b]$ shares the well-known dynamical metastability valid for the classical parabolic case. In particular, using the “energy approach” proposed by Bronsard and Kohn [On the slowness of phase boundary motion in one space dimension, Comm. Pure Appl. Math. 43 (1990) 983–997], if $\epsilon \ll 1$ is the diffusion coefficient, we show that in a time scale of order ${\epsilon }^{-k}$ nothing happens and the solution maintains the same number of transitions of its initial datum $u_0$. The novelty consists mainly in the role of the initial velocity $u_1$, which may create or eliminate transitions in later times. Numerical experiments are also provided in the particular case of the Allen–Cahn equation with relaxation.

We investigate a one-dimensional nonlinear wave system which is the Euler–Lagrange equations of a variational principle modeling a type of cholesteric liquid crystals. Due to the chiral effect, the Lagrangian density of the variational principle includes linear terms as well as quadratic terms in derivatives of the field, which is different from the nematic liquid crystals. Moreover, the linear terms and zero term in derivatives of the field might cause the energy to be negative in some cases. By using the method of energy-dependent coordinates and the Young measure theory, we establish the global existence of weak solutions to its Cauchy problem under some conditions on the coefficients.

We consider a two-dimensional (2D) steady jet generated by gas streaming in parallel supersonic flow out of a duct and into the atmosphere. When the gas is a van der Waals gas, at the corners at the exit of the duct, the parallel supersonic flow expands symmetrically into centered simple waves, jump–fan (JF) composite waves, fan–jump composite waves, or fan–jump–fan (FJF) composite waves. This paper studies the interaction of the symmetric centered FJF composite waves in the jet. The interaction of the FJF composite waves is more involved in comparison to our earlier work on the interaction of the symmetric centered JF composite waves, since the flow in the FJF composite wave interaction zone is rotational. To construct the flow in the interaction zone, we consider a Goursat-type boundary value problem with discontinuous boundary data associated with the 2D and isentropic steady Euler equations. The existence of a global piecewise smooth solution to this Goursat problem is obtained by a new method based on characteristics.

We study a quasilinear nonlocal Benney system and establish the existence and uniqueness of strong local in time solutions to the corresponding Cauchy problem. We also show, under certain conditions, the blow-up of such solutions in finite time. Furthermore, we prove the existence of global weak solutions and exhibit bound-state solutions to this system.

We consider the quantum Zakharov system in one spatial dimension and establish a local well-posedness theory when the initial data of the electric field and the deviation of the ion density lie in a Sobolev space with suitable regularity. As the quantum parameter approaches zero, we formally recover a classical result by Ginibre, Tsutsumi, and Velo. We also improve their result concerning the Zakharov system and a result by Jiang, Lin, and Shao concerning the quantum Zakharov system.