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In this paper, we investigate the well-posedness and large time behavior of solutions to the 3D incompressible magneto-micropolar equations. By virtue of the $ L_p-L_q $ L p − L q estimate obtained through the spectral decomposition of the linearized magneto-micropolar equations, we show the existence and uniqueness of small $ L_3 $ L 3 -strong solutions of the equations with small initial data. Then basing on this result, we derive sharp time decay estimates of the $ L_3 $ L 3 -strong solutions. [ABSTRACT FROM AUTHOR]

The zero-relaxation limit of Euler–Poisson systems for ion dynamics in a slow time scale leads to drift-diffusion equations. This fact was justified in previous works. This paper concerns global error estimates between solutions of Euler–Poisson systems and that of drift-diffusion equations. In the proof, we use Sobolev inequalities and employ a multiplier related to the electric potential. [ABSTRACT FROM AUTHOR]

In this paper, we shall investigate the large-time behavior of solutions to the Cauchy problem for the three-dimensional bipolar isentropic Euler–Poisson equations with time-dependent damping effects $ -\frac {\mu }{(1+t)^{\lambda }}n_iu_i(i=1, 2) $ − μ (1 + t) λ n i u i (i = 1 , 2) for $ -1 − 1 < λ < 1 , μ > 0. Here, we consider a more general case that the two pressure functions are different and the doping profile is non-zero. Under the assumption that the initial data are close to the constant equilibrium states, we show that the smooth solutions to the Cauchy problem exist uniquely and globally. The algebraic time-decay rates of the solutions toward the constant equilibrium states are also obtained. The main ingredient of the proof is the time-weighted energy method with artfully chosen time-weighted functions. [ABSTRACT FROM AUTHOR]

This paper is concerned with the incompressible limit of the non-isentropic ideal magnetohydrodynamic equations with periodic boundary conditions and general initial data. Weak convergence results of the smooth solutions are established by filtering method in the torus. [ABSTRACT FROM AUTHOR]

Of concern is a problem of strain gradient porous elastic theory with nonlinear damping terms, whose constitutive equations contain higher-order derivatives of the displacement in the basic postulates. The paper is based on the theory of 'consistency' due to Aouadi et al. [J. Therm. Stress. 43(2)(2020), 191–209] and Ieşan [American Institute of Physics, Conference Proceedings, 1329 (2011), 130–149], and contains four results. We firstly show that the system is global well posed by using maximal monotone operator. The second main result is the existence of global attractors which is proved by the method developed by Chueshov and Lasiecka [Long-time behavior of second order evolution equations with nonlinear damping. Mem. Amer. Math. Soc. vol. 195, no. 912, Providence, 2008; Von Karman evolution equations: well-posedness and long-time dynamics. Springer Monographs in Mathematics, Springer, New York, 2010]. By showing the system is gradient and asymptotically smooth, we establish the existence of global attractors with finite fractal dimension via a stabilizability inequality. Then we study the continuity of global attractors regarding the parameter in a residual dense set. The above results allow the damping terms with polynomial growth. Finally we discuss the exponential decay and global boundedness to the linear case of damping terms of the system. The assumption of equal-speed wave propagations is not needed for all of results obtained. [ABSTRACT FROM AUTHOR]

In this paper, we first show the nondegeneracy and asymptotic behavior of ground states for the nonlinear fourth-order Schrödinger equation with mixed dispersion: \[ \delta\Delta^{2}u-\Delta u +u=|u|^{2\sigma}u, \quad u\in H^{2}(\mathbb{R}^{N}), \] δ Δ 2 u − Δu + u = | u | 2 σ u , u ∈ H 2 (R N) , where $ \delta \gt 0 $ δ > 0 is sufficiently small, $ 0 \lt \sigma \lt \frac {2}{(N-2)_+} $ 0 < σ < 2 (N − 2) + , $ \frac {2}{(N-2)_+}=\frac {2}{N-2} $ 2 (N − 2) + = 2 N − 2 for $ N\geq ~3 $ N ≥ 3 and $ \frac {2}{(N-2)_+}=+\infty $ 2 (N − 2) + = + ∞ for N=2,3. This work extends some results in Bonheure, Casteras, Dos Santos, and Nascimento [Orbitally stable standing waves of a mixed dispersion nonlinear Schrödinger equation. SIAM J Math Anal. 2018;50:5027–5071]. Next, suppose $ P(x) $ P (x) and $ Q(x) $ Q (x) are two positive, radial and continuous functions satisfying that as $ r=|x|\rightarrow +\infty $ r = | x | → + ∞ , \[ P(r)=1+\frac{a_1}{r^{m_1}} +O\left(\frac{1}{r^{m_1+\theta_1}}\right),\quad Q(r)=1+\frac{a_2}{r^{m_2}} +O\left(\frac{1}{r^{m_2+\theta_2}}\right), \] P (r) = 1 + a 1 r m 1 + O (1 r m 1 + θ 1 ) , Q (r) = 1 + a 2 r m 2 + O (1 r m 2 + θ 2 ) , where $ a_1, a_2\in \mathbb {R} $ a 1 , a 2 ∈ R , $ m_1, m_2 \gt 1 $ m 1 , m 2 > 1 , $ \theta _1, \theta _2 \gt 0 $ θ 1 , θ 2 > 0. We use the Lyapunov–Schmidt reduction method developed by Wei and Yan [Infinitely many positive solutions for the nonlinear Schrödinger equations in RN. Calc Var. 2010;37:423–439] to construct infinitely many nonradial positive and sign-changing solutions with arbitrary large energy for the following equation: \[ \delta\Delta^{2}u-\Delta u +P(x)u=Q(x)|u|^{2\sigma}u, \quad u\in H^{2}(\mathbb{R}^{N}). \] δ Δ 2 u − Δu + P (x) u = Q (x) | u | 2 σ u , u ∈ H 2 (R N). [ABSTRACT FROM AUTHOR]

In this paper, we study the global strong solutions to the three-dimensional (3D) heat-conducting incompressible Navier–Stokes equations with density-temperature-dependent viscosity and heat-conducting coefficients in $ \mathbb {R}^3 $ R 3 . By using the t-weighted a priori estimates, we prove the global existence and exponential decay-in-time rates of strong solutions to the Cauchy problem when the $ L^{{3}/{2}} $ L 3 / 2 -norm of the initial density is suitably small. It should be noted that the velocity and absolute temperature can be large initially, and the initial density contains vacuum case. [ABSTRACT FROM AUTHOR]

This paper is concerned with the initial-boundary value problem for a logarithmic Lamé system with a time delay in a bounded domain. We prove the well-posedness of the system by utilizing the semigroup theory. Then, we prove the existence of global solutions by using the well-depth method. In addition, we establish an exponential stability decay result under appropriate assumptions on the weight of the time delay and that of frictional damping. [ABSTRACT FROM AUTHOR]

A reaction-diffusion-ODE system of stoichiometric producer-grazer type is considered in this paper. Since the system has nonsmooth nonlinearity, it is shown that the system has new dynamics different from the smooth case. We construct various types of discontinuous steady states and investigate their asymptotic behavior. In addition, the steady-state bifurcation near a constant solution is studied by treating the diffusion coefficient as a bifurcation parameter and the existence of Hopf bifurcation is derived. Our results cover the case where the number of positive equilibria of the kinetic system (i.e. without diffusion) changes from one to three in the spatial interval. Finally, some numerical simulations are given to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

This paper is concerned with the long-term behaviour of 3D double time-delayed magnetohydrodynamics (MHD)-Voight equations with non-autonomous forcing term. First, we establish the existence, forward compactness and forward stability of pullback attractors by the forward equi-continuity and forward pullback asymptotic compactness of solutions. We then study the upper semi-convergence of pullback attractors as the time parameter tends to positive infinity. Since the solution of this equation has no high regularity, we use the spectrum decomposition technique to prove the forward pointwise pullback asymptotic compactness of solutions. [ABSTRACT FROM AUTHOR]