Large Time Behavior Of Solutions Research Articles

Large Time Behavior of Solutions to the 3D Rotating Navier–Stokes Equations

We consider the large time behavior of the solutions for the initial value problem of the Navier–Stokes equations with the Coriolis force in the three-dimensional whole space. We show the L^p temporal decay estimates with the dispersion effect of the Coriolis force for the global solutions. Moreover, we prove the large time asymptotic expansion of the solutions behaving like the first-order spatial derivatives of the integral kernel of the corresponding linear solution.

Nonlinear evolution equation associated with hypergraph Laplacian

Let be a finite set, be a set of hyperedges, and be an edge weight. On the (wighted) hypergraph , we can define a multivalued nonlinear operator ( ) as the subdifferential of a convex function on , which is called “hypergraph ‐Laplacian.” In this article, we first introduce an inequality for this operator , which resembles the Poincaré–Wirtinger inequality in PDEs. Next, we consider an ordinary differential equation on governed by , which is referred as “heat” equation on the graph and used to study the geometric structure of the hypergraph in recent researches. With the aid of the Poincaré–Wirtinger type inequality, we can discuss the existence and the large time behavior of solutions to the ODE by procedures similar to those for the standard heat equation in PDEs with the zero Neumann boundary condition.

Decay property of solutions to the wave equation with space‐dependent damping, absorbing nonlinearity, and polynomially decaying data

We study the large time behavior of solutions to the semilinear wave equation with space‐dependent damping and absorbing nonlinearity in the whole space or exterior domains. Our result shows how the amplitude of the damping coefficient, the power of the nonlinearity, and the decay rate of the initial data at the spatial infinity determine the decay rates of the energy and the ‐norm of the solution. In the appendix, we also give a survey of basic results on the local and global existence of solutions and the properties of weight functions used in the energy method.

Large time behavior of solutions to a reduced gravity two and a half layer model with damping

Large time behavior of solutions to a reduced gravity two and a half layer model with damping

On asymptotics of solutions for superdiffusion and subdiffusion equations with the Riemann-Liouville fractional derivative

<abstract><p>In the present paper, we focus on the study of the asymptotic behaviors of solutions for the Cauchy problem of time-space fractional superdiffusion and subdiffusion equations with integral initial conditions, where the Riemann-Liouville derivative is used in the temporal direction and the integral fractional Laplacian is applied in the spatial variables. The fundamental solutions of the considered equations, which can be represented in terms of the Fox $ H $-function, are constructed and investigated by using asymptotic expansions of the Fox $ H $-function. Then, we obtain the asymptotic behaviors of solutions in the sense of $ L^{p}(\mathbb{R}^{d}) $ and $ L^{p, \infty}(\mathbb{R}^{d}) $ norms, where Young's inequality for convolution plays a very important role. Finally, gradient estimates and large time behaviors of solutions are also provided. In particular, we derive the optimal $ L^{2} $- decay estimate for the subdiffusion equation.</p></abstract>

Asymptotic stability of a composite wave to the one-dimensional compressible Navier-Stokes equations for a reacting mixture

This paper is concerned with the large time behavior of solutions for the Cauchy problem to the compressible Navier-Stokes equations for a reacting mixture with zero viscosity in one dimension. If the corresponding Riemann problem for the compressible Euler system admits the solutions consisting of rarefaction waves and contact discontinuity, it is shown that the composition of two rarefaction waves with a viscous contact wave is asymptotically stable, while the strength of the composite wave and the initial perturbation are suitably small.

Large time behavior of solutions for a PDE model for compressible elastic curve

Large time behavior of solutions for a PDE model for compressible elastic curve

The continuous Redner-Ben-Avraham-Kahng coagulation system: Well-posedness and asymptotic behaviour

This paper examines the existence of solutions to the continuous Redner-Ben-Avraham-Kahng coagulation system under specific growth conditions on unbounded coagulation kernels at infinity. Moreover, questions related to uniqueness and continuous dependence on the data are also addressed under additional restrictions. Finally, the large-time behaviour of solutions is also investigated.

Large time behavior of solutions to the 3D anisotropic Navier–Stokes equation

We consider the large time behavior of the solution to the 3D Navier–Stokes equation with horizontal viscosity Δhu=∂12u+∂22u and show that the Lp decay rate of the horizontal components of the velocity field coincides to that of the 2D heat kernel, while the vertical component decays like the 3D heat kernel. Moreover, we consider the asymptotic expansion of the solution and find that a portion of the nonlinear term affect the leading term of the horizontal components of the velocity field, whereas the leading term of the vertical component is given by only the linear solution.

Global existence and decay estimates of solutions to the MHD–Boussinesq system with stratification effects* *This work was supported by National Natural Science Foundation of China, NSFC (No.: 11926316, 11531010, 12071391).

In this paper, we consider the inviscid Boussinesq system under the presence of magnetic field in porous media. We firstly proved the global well-posedness and large time behaviour of solutions in the whole space . Precisely, when the H 3 norm of initial data is small, but the higher order derivatives can be arbitrary large, the system is globally well-posed by pure energy method. Moreover, by a set of mature negative Sobolev and Besov space interpolation methods, the L p − L 2 (1 ⩽ p ⩽ 2) type of the optimal time decay rates are obtained without any smallness assumption on the L p norm of the initial data. At last, we derive a global weak solution with large initial data and give an explicit decay rate of the solution in . Our results mathematically explain the stable phenomenon of the system with stratification effects.

Global well-posedness and large time behavior of epitaxy thin film growth model

We consider the global well-posedness and large time behavior of solutions for epitaxy thin film growth model in mathbb{R}^{d} with the dimensional dgeq 3. First, using the pure energy method and a standard continuity argument, we prove that there exists a unique global strong solution under the condition that the initial data is sufficiently small. Moreover, we also establish the suitable negative Sobolev norm estimates and obtain the optimal decay rates of the higher-order spatial derivatives of the strong solution.

Large time behavior of solutions to a class of hyperbolic inequalities with mixed nonlinearities

We investigate the large time behavior of solutions to a class of inhomogeneous hyperbolic inequalities involving combined nonlinearities of the form |u|p+1Γ(σ)∫0t(t−s)σ−1|∇u(s,x)|qds+w(x), where p,q>1 and σ>0. We show that, if w has a positive average, then the considered class admits no global weak solution. Our approach is based on nonlinear capacity estimates specifically adapted to the nonlocal setting of our problem.

Large time behavior of solution to a fully parabolic chemotaxis system with singular sensitivity and logistic source

This paper presents the large time behavior of solution to the fully parabolic chemotaxis system with singular sensitivity and logistic source ut=∇⋅(D(u)∇u)−χ∇⋅(uvκ∇v)+μu−μu2,x∈Ω,t>0,vt=Δv−v+u,x∈Ω,t>0,with homogeneous Neumann boundary condition in a convex smooth bounded domain Ω⊂Rn, n≥2, where χ>0, μ>0 and κ∈(0,12)∪(12,1), D(u) is supposed to satisfy the following property D(u)≥(u+1)αwithα>0.One can find a positive constant m∗ such that ∫Ωu≥m∗for allt≥0.Apart from that, it is shown that the solution is globally bounded. Furthermore, it is asserted that the solution exponentially converges to the steady state (1,1) as t→∞.

Large time behavior of ODE type solutions to higher-order semilinear parabolic equations with small initial data

Let u be a solution to the Cauchy problem for semilinear parabolic equations(Pm){∂tu+(−Δ)mu=|u|αinRN×(0,∞),u(x,0)=λ+φ(x)>0inRN, where m=1,2,⋯,0<α<1,N≥1,λ>0,φ∈BC(RN)∩Lr(RN) with 1≤r<∞. In the case of m=1, by the comparison principle, one can easily show that the solution u to problem (Pm) behaves like a positive solution to ODE ζ′=ζα in (0,∞) as t→∞. In the case of m≥2, because of the lack of the comparison principle, it is not obvious that the solution u to problem (Pm) behaves like a solution to the ODE as t→∞. In this paper, by imposing a smallness condition‖φ‖L∞(RN)<c with a certain constant c dependent on m,α and λ, we prove that the solution u behaves like a solution to the ODE as t→∞. Furthermore, we obtain the precise description of the large time behavior of the solution u and reveal the relationship between the diffusion effect (Pm) has.

Large time behavior of solutions to the Cauchy problem for the BBM–Burgers equation

We consider the large time behavior of the solutions to the Cauchy problem for the BBM–Burgers equation. We prove that the solution to this problem goes to the self-similar solution to the Burgers equation called the nonlinear diffusion wave. Moreover, we construct the appropriate second asymptotic profiles of the solutions depending on the initial data. Based on that discussion, we investigate the effect of the initial data on the large time behavior of the solution, and derive the optimal asymptotic rate to the nonlinear diffusion wave. Especially, the important point of this study is that the second asymptotic profiles of the solutions with slowly decaying data, whose case has not been studied, are obtained.

Large time behavior of solutions to the critical dissipative nonlinear Schrödinger equation with large data

Large time behavior of solutions to the critical dissipative nonlinear Schrödinger equation with large data

Large time behavior of solutions to a system of coupled nonlinear oscillators via a generalized form of Schauder-Tychonoff fixed point theorem

Large time behavior of solutions to a system of coupled nonlinear oscillators via a generalized form of Schauder-Tychonoff fixed point theorem

Large time behavior of solutions to the inviscid liquid‐gas two‐phase flow model

The purpose of this paper is to gain some insight into the existence theory and large time behavior of the inviscid liquid‐gas two‐phase flow model in . Main focus is on the influences of the damping on the qualitative behavior of solutions of the corresponding Cauchy problem under only the smallness assumption on ‐norm of the initial perturbation. The energy method combined with the low frequency and high frequency decomposition is used to obtain the desired decay of the solution and hence global existence. As a byproduct, ‐ decay rates are obtained for the solution and its derivatives. This result implies that the friction effect of the damping contributes to enhance the decay rate of the velocity, even if there is no the additional ‐norm boundedness condition of the initial perturbation.

Large-Time Behavior of Solutions in the Critical Spaces for the Non-isentropic Compressible Navier–Stokes Equations with Capillarity

Large-Time Behavior of Solutions in the Critical Spaces for the Non-isentropic Compressible Navier–Stokes Equations with Capillarity

Global boundedness and large time behavior of solutions to a chemotaxis system with flux limitation

This paper deals with the following cross-diffusion system{ut=∇⋅((u+1)m−1∇u)−∇⋅(u∇v(1+|∇v|2)α),x∈Ω,t>0,0=Δv−v+u,x∈Ω,t>0, in a bounded domain Ω⊂Rn, n≥2 with smooth boundary. In this framework, it is shown that the problem possesses a unique global bounded classical solution under the condition that α>2n−2−mn2(n−1) and m≥1. Moreover, it is asserted that the corresponding solution exponentially converges to the constant stationary solution (u0‾,u0‾) when the initial data u0 is sufficiently small, where u0‾:=∫Ωu0|Ω|.