This article studies the dynamical behaviour of classical solutions of a hyperbolic system of balance laws, derived from a chemotaxis model with logarithmic sensitivity, with time-dependent boundary conditions. It is shown that under suitable assumptions on the boundary data, solutions starting in the $H^2$-space exist globally in time and the differences between the solutions and their corresponding boundary data converge to zero as time goes to infinity. There is no smallness restriction on the magnitude of the initial perturbations. Moreover, numerical simulations show that the assumptions on the boundary data are necessary for the above-mentioned results to hold true. In addition, numerical results indicate that the solutions converge asymptotically to time-periodic states if the boundary data are time-periodic.

This paper focuses on the vanishing limit problem for the three-dimensional incompressible Phan-Thien–Tanner (PTT) system, which is commonly used to describe the dynamic properties of polymeric fluids. Our purpose is to show the relation of the PTT system to the well-known Oldroyd-B system (with or without damping mechanism). The suitable a priori estimates and global existence of strong solutions are established for the PTT system with small initial data. Taking advantage of uniform energy and decay estimates for the PTT system with respect to time $t$ and coefficients $a$ and $b$, then allows us to justify in particular the vanishing limit for all time. More precisely, we prove that the solution $(u,\,\tau )$ of PTT system with $0\leq b\leq Ca$ converges globally in time to some limit $(\widetilde {u},\,\widetilde {\tau })$ in a suitable Sobolev space when $a$ and $b$ go to zero simultaneously (or, only $b$ goes to zero). We may check that $(\widetilde {u},\,\widetilde {\tau })$ is indeed a global solution of the corresponding Oldroyd-B system. In addition, a rate of convergence involving explicit norm will be obtained. As a byproduct, similar results are also true for the local a priori estimates in large norm.

We study the small data global well-posedness and time-decay rates of solutions to the Cauchy problem for three-dimensional compressible Navier–Stokes–Allen–Cahn equations via a refined pure energy method. In particular, the optimal decay rates of the higher-order spatial derivatives of the solution are obtained, the $\dot {H}^{-s}$($0\leq s<\frac {3}{2}$) negative Sobolev norms is shown to be preserved along time evolution and enhance the decay rates.

We deal with an initial boundary value problem of nonhomogeneous Boussinesq equations for magnetohydrodynamics convection in two-dimensional domains. We prove that there is a unique global strong solution. Moreover, we show that the temperature converges exponentially to zero in H1 as time goes to infinity. In particular, the initial data can be arbitrarily large and vacuum is allowed. Our analysis relies on energy method and a lemma of Desjardins (Arch. Rational Mech. Anal. 137:135–158, 1997).

We show the incompressible Navier–Stokes–Maxwell system with solenoidal Ohm's law can be derived from the two-fluid incompressible Navier–Stokes–Maxwell system when the momentum transfer coefficient tends to zero. The strategy is based on the decay and dissipative properties of the electromagnetic field.

We consider the Cauchy problem for the nonlinear Schrödinger equation on the whole space. After introducing a weaker concept of finite speed of propagation, we show that the concatenation of initial data gives rise to solutions whose time of existence increases as one translates one of the initial data. Moreover, we show that, given global decaying solutions with initial data u0, v0, if |y| is large, then the concatenated initial data u0 + v0(· − y) gives rise to globally decaying solutions.

We study the Cauchy problem for the generalised fourth-order Schrödinger equation for data u0 in critical Sobolev spaces . With small initial data we obtain global well-posedness results. Our proof relies heavily on the method developed by Kenig et al. [‘Well-posedness and scattering results for the generalised Korteweg–de Vries equation via the contraction principle’, Commun. Pure Appl. Math.46 (1993), 527–620].

We prove that the modified Benjamin–Ono–Burgers equation is globally well-posed in Hs for s>0. Moreover, we show that the solution of the modified Benjamin–Ono–Burgers equation converges to that of the modified Benjamin–Ono equation in the natural space C([0,T];Hs), s≥1/2, as the dissipative coefficient ϵ goes to zero, provided that the L2 norm of the initial data is sufficiently small.