Large time behavior of solutions to the Cauchy problem for one-dimensional thermoelastic system with dissipation

We study the large time behavior of the weak solutions to a one-phase moving sharp-interface PDE system describing the aggressive penetration of gaseous carbon dioxide in unsaturated concrete. The key of the proof is a global uniform estimate for solutions obtained by using the maximum principle. The analysis reported here relies on the global existence and uniqueness of solutions that we have proved previously.

We investigate the asymptotic behavior of a solution of a Hamilton–Jacobi equation defined on a general metric space. Our results include general stability property and large time behavior of the solution. We argue from the viewpoint of viscosity solutions theory and adopt the notion of solution recently introduced by Gangbo and Święch to the Hamilton–Jacobi equation on a geodesic space.

We study the large time behavior of the solutions for the Cauchy problem, ∂ t u = Δu + a(x, t)u in ℝ N × (0, ∞), u (x, 0) = Q(x) in ℝ N , where Q ∈ L 1 (ℝ N , (1+|x| K ) dx) with K ≥ 0 and ∥a(t)∥ L ∞ (ℝ N ) = O(t -A ) as t → ∞ for some A > 1. In this paper we classify the decay rate of the solutions and give the precise estimates on the difference between the solutions and their asymptotic profiles. Furthermore, as an application, we discuss the large time behavior of the global solutions for the semilinear heat equation, ∂ t u = Δu + λ|u| p-1 u, where λ ∈ ℝ and p > 1.

The present paper is dedicated to the optimal time-decay estimates of global strong solutions near constant equilibrium (away from vacuum) to the compressible magnetohydrodynamic (MHD) equations in the critical Besov spaces. In which we claim a new low-frequency assumption that plays a key role in the large-time behavior of solutions. Precisely, we exhibit that if the low frequencies of initial data belong to some Besov space with (), then the norm (the slightly stronger norm in fact) of strong solutions has the optimal decay ( if ) for , which improve the results of [Shi WX, Xu J. Large-time behavior of strong solutions to the compressible magnetohydrodynamic system in the critical framework. J Hyperbol Differ Equ. 2018;15:259–290]. The proof mainly depends on a sharp time-weighted energy estimates in light of low and high frequencies for the solutions. As a by-product, those optimal time-decay rates of - type are also captured in the critical framework.

Large time behavior for convection-diffusion equations in R N

The present article is devoted to the study of global solution and large time behaviour of solution for the isentropic compressible Euler system with source terms in $\mathbb {R}^d$ , $d\geq 1$ , which extends and improves the results obtained by Sideris et al. in ‘T.C. Sideris, B. Thomases, D.H. Wang, Long time behavior of solutions to the 3D compressible Euler equations with damping, Comm. Partial Differential Equations 28 (2003) 795–816’. We first establish the existence and uniqueness of global smooth solution provided the initial datum is sufficiently small, which tells us that the damping terms can prevent the development of singularity in small amplitude. Next, under the additional smallness assumption, the large time behaviour of solution is investigated, we only obtain the algebra decay of solution besides the $L^2$ -norm of $\nabla u$ is exponential decay.

The compressible non-isentropic Navier-Stokes-Maxwell system is investigated in $\mathbb{R}^3$ and the global existence and large time behavior of solutions are established by pure energy method provided the initial perturbation around a constant state is small enough. We first construct the global unique solution under the assumption that the $H^3$ norm of the initial data is small, but the higher order derivatives can be arbitrarily large. If further the initial data belongs to $\dot{H}^{-s}$ ($0≤ s<3/2$) or $\dot{B}_{2,∞}^{-s}$ ($0< s≤3/2$), by a regularity interpolation trick, we obtain the various decay rates of the solution and its higher order derivatives. As an immediate byproduct, the $L^p$-$L^2$ $(1≤ p≤ 2)$ type of the decay rates follows without requiring that the $L^p$ norm of initial data is small.

Decay estimate of solutions to the coupled chemotaxis–fluid equations in [formula omitted

Soliton resolution and large time behavior of solutions to the Cauchy problem for the Novikov equation with a nonzero background

Asymptotic profiles for convection–diffusion equations with variable diffusion

Large time behavior of solutions to the compressible Navier-Stokes equation around a given constant state is considered in an infinite layer ${\bf R}^{n-1}\times (0,a)$, $n\geq2$, under the no slip boundary condition for the velocity. The $L^p$ decay estimates of the solution are established for all $1\leq p\leq \infty$. It is also shown that the time-asymptotic leading part of the solution is given by a function satisfying the $n-1$ dimensional heat equation. The proof is given by combining a weighted energy method with time-weight functions and the decay estimates for the associated linearized semigroup

We consider the vorticity formulation of the two-dimensional viscous Camassa–Holm equations in the whole space. We establish global existence for solutions corresponding to initial data in $L^1$ and describe the large time behavior of solutions with sufficiently small and localized initial data. We calculate the rate at which such solutions approach an “unfiltered” Oseen vortex by computing the rate at which the solution of a scaled vorticity problem approaches the solution to a corresponding linearized equation.

There have been extensive studies on the large time behavior of solutions to systems on gas motions, such as the Navier-Stokes equations and the Boltzmann equation. Recently, an approach is introduced by combining the energy method and the spectral analysis to the study of the optimal rates of convergence to the asymptotic profiles. In this paper, we will first illustrate this method by using some simple model and then we will present some recent results on the Navier-Stokes equations and the Boltzmann equation. Precisely, we prove the stability of the non-trivial steady state for the Navier-Stokes equations with potential forces and also obtain the optimal rate of convergence of solutions toward the steady state. The same issue was also studied for the Boltzmann equation in the presence of the general time-space dependent forces. It is expected that this approach can also be applied to other dissipative systems in fluid dynamics and kinetic models such as the model system of radiating gas and the Vlasov-Poisson-Boltzmann system.

This paper is concerned with the heat equation with a reaction source of spatio-temporal delay. Special attention is paid to the determination of critical exponents used to describe large time behaviour of solutions of the Cauchy problem.

In this paper, we consider large time behaviour of solutions to the 3D Navier–Stokes equations with damping term $$|u|^{\beta -1}u$$ ( $$\beta \ge 1$$ ). For $$\beta >\frac{7}{3}$$ , we derive decay rates of the $$L^2$$ -norm of the solutions. By using Fourier splitting method, we also prove that the solutions are asymptotically equivalent to the solutions of the classic 3D Navier–Stokes equations.