Asymptotic Behavior Of Weak Solutions Research Articles

Regularity and attractors for the three‐dimensional generalized Boussinesq system

We study the three‐dimensional Boussinesq system with dissipation and diffusion generalized in terms of fractional Laplacians. First, we study the existence, uniqueness, and regularity of global weak solutions. Then, we investigate the asymptotic behavior of weak solutions via attractors. Since the system might not always have regular solutions, we use a new framework developed by Cheskidov and Lu which is called evolutionary system to obtain various attractors and their properties.

Gradient and Parameter Dependent Dirichlet (p(x),q(x))-Laplace Type Problem

We analyze a Dirichlet (p(x),μq(x))-Laplace problem. For a gradient dependent nonlinearity of Carathéodory type, we discuss the existence, uniqueness and asymptotic behavior of weak solutions, as the parameter μ varies on the non-negative real axis. The results are obtained by applying the properties of pseudomonotone operators, jointly with certain a priori estimates.

Asymptotic Behavior of Weak Solutions to the Inhomogeneous Navier–Stokes Equations

Asymptotic Behavior of Weak Solutions to the Inhomogeneous Navier–Stokes Equations

On a viscoelastic heat equation with logarithmic nonlinearity

This work deals with the following viscoelastic heat equations with logarithmic nonlinearity u t − Δ u + ∫ 0 t g ( t − s ) Δ u ( s ) d s = | u | p − 2 u ln ⁡ | u | . In this paper, we show the effects of the viscoelastic term and the logarithmic nonlinearity to the asymptotic behavior of weak solutions. Our results extend the results of Peng and Zhou [Appl. Anal. 100(2021), 2804–2824] and Messaoudi [Nonlinear Differ. Equ. Appl. 64(2005), 351–356].

Asymptotic Behavior of Weak Solutions of Nonisothermal Flow of Herschel–Bulkley Fluid to Free Boundary

In this manuscript, the behavior of a Herschel–Bulkley fluid has been discussed in a thin layer in ℝ3 associated with a nonlinear stationary, nonisothermal, and incompressible model. Furthermore, the limit problem has been considered, and the studied problem in Ωε is transformed into another problem defined in Ωε without the parameter Ωε (ε is the parameter representing the thickness of the layer tend to zero is studied). We also investigated the convergence of the unknowns which are the velocity, pressure, and the temperature of the fluid. In addition, we established the limit problem and the specific Reynolds equation.

Global existence, asymptotic behavior, and regularity of solutions for time–space fractional Rosenau equations

In this paper, we investigate initial boundary value problems for the following time–space fractional Rosenau equation with the Caputo time fractional derivatives and the spectral fractional Laplacian operators. To the best of our knowledge, there are few results concerning the time–space fractional Rosenau equations. The main difficulties are the nonlocal effects generated by the operators and (− Δ)γ. First, we establish some rough and rigorous decay estimates of weak solutions to the corresponding linear equations, respectively. Based on the decay estimates, under small initial value condition, we prove the global existence and asymptotic behavior of weak solutions in the time‐weighted Sobolev spaces by the contraction mapping principle. Furthermore, we discuss the regularity of weak solutions when initial value data are strengthened from to .

Precise decay rates of global solutions and an explicit upper bound of the blow‐up time to a pseudo‐parabolic equation

This paper deals with a pseudo‐parabolic equation with nonstandard growth conditions. Some results on global existence, blow‐up, and asymptotic behavior of weak solutions were obtained. The purpose of this paper is to give results to some unsolved questions. First, the precise decay rates for global solutions are derived. Second, an explicit upper bound of the blow‐up time is obtained when the initial energy is low.

Local versus nonlocal elliptic equations: short-long range field interactions

Abstract In this paper we study a class of one-parameter family of elliptic equations which combines local and nonlocal operators, namely the Laplacian and the fractional Laplacian. We analyze spectral properties, establish the validity of the maximum principle, prove existence, nonexistence, symmetry and regularity results for weak solutions. The asymptotic behavior of weak solutions as the coupling parameter vanishes (which turns the problem into a purely nonlocal one) or goes to infinity (reducing the problem to the classical semilinear Laplace equation) is also investigated.

Algebraic [formula omitted]-decay of weak solutions to the magneto-hydrodynamic equations

We consider the asymptotic behavior of weak solutions to the incompressible magneto-hydrodynamics (MHD) equations in the whole space Rn, n≥2. The algebraic decay of the total energy of weak solutions to Cauchy problem of MHD equations is given by establishing an important inequality on weak solutions of MHD equations.

Asymptotic behavior of weak solutions to the damped Navier-Stokes equations

Whether or not weak solutions of classical Navier-Stokes equations decay to zero in L2 as time tends to infinity was posed by Leray in his pioneering paper [12,13] and has been well solved (see [15] for instance). This paper examines the three dimensional Navier-Stokes equations with damping term |u|β−1u and proves that the weak solutions decay to zero in L2 as time tends to infinity for any β≥1, by developing local-in-space estimate for weak solutions.

On a Boltzmann Equation for Compton Scattering from Non relativistic Electrons at Low Density

A Boltzmann equation, used to describe the evolution of the density function of a gas of photons interacting by Compton scattering with electrons at low density and non relativistic equilibrium, is considered. A truncation of the very singular redistribution function is introduced and justified. The existence of weak solutions is proved for a large set of initial data. A simplified equation, where only the quadratic terms are kept and that appears at very low temperature of the electron gas, for small values of the photon's energies, is also studied. The existence of weak solutions, and also of more regular solutions that are very flat near the origin, is proved. The long time asymptotic behavior of weak solutions of the simplified equation is described.

Asymptotic behavior for a fourth-order parabolic equation modeling thin film growth

The main goal of this work is to study a fourth-order parabolic equation with nonstandard growth conditions. For non-positive initial energy J(u0)≤0, by using the concavity method, we establish that weak solutions blow up in finite time. On the other hand, we also derive the blow-up time estimate and the asymptotic behavior of weak solutions.

Weakly regular fluid flows with bounded variation on the domain of outer communication of a Schwarzschild black hole spacetime. II

We study the evolution of relativistic fluids in presence of shock waves on a curved background and, more precisely, we propose here a rather complete treatment of the Cauchy problem for the relativistic Burgers equation posed on the domain of outer communication of a Schwarzschild black hole spacetime. This spacetime contains a horizon (on which the characteristic speed of our Burgers equation vanishes) and an asymptotically flat end (around which our equation is asymptotic to the standard Burgers equation). We establish a global existence theory for weak solutions, when the initial data have (suitably weighted) bounded total variation. Our proof relies on a Glimm method which preserves the class of static solutions and uses as a building block the solutions of the generalized Riemann problem associated with static solutions. In addition, we investigate the long-time asymptotic behavior of weak solutions, when the initial data is a perturbation of static state solutions separated by a jump discontinuity.

Asymptotic Behavior of Weak Solutions to the Generalized Nonlinear Partial Differential Equation Model

This paper investigates the asymptotic behavior of weak solutions to the generalized nonlinear partial differential equation model. It is proved that every perturbed weak solution of the perturbed generalized nonlinear partial differential equations asymptotically converges to the solution of the original system under the large perturbation.

Decay rate estimates for a class of quasilinear hyperbolic equations with damping terms involving p-Laplacian

In this paper, we are concerned with the asymptotic behaviour of weak solutions to the initial boundary value problem for a class of quasilinear hyperbolic equations with damping terms involving p-Laplacian. By using the multiplier methods, we investigate the stability of weak solutions to the initial boundary value problem and obtain explicit decay rate estimation depending on strain-caused stress term and damping terms.

Global attractor for the regularized Bénard problem

In this paper, we study the asymptotic behaviour of weak solutions for the regularized Bénard problem. We establish the global existence and uniqueness of weak solutions of this problem and give a proof for the existence of global attractor in the three-dimensional case for this system.

Korteweg–de Vries–Burgers system in [formula omitted

We study Cauchy problem in RN for the Korteweg–de Vries–Burgers system. Parabolic regularization technique is used to prove its global in time solvability. The regularization effect of the Laplacian term is observed for the viscous solutions constructed in the paper. Asymptotic behavior of weak solutions is discussed next in the language of the global attractors.

On one-dimensional compressible Navier-Stokes equations with degenerate viscosity and constant state at far fields

In this paper, we are concerned with the Cauchy problem for one-dimensional compressible isentropic Navier-Stokes equations with density-dependent viscosity μ(ρ) = ρα(α > 0) and pressure P(ρ) = ργ (γ > 1). We will establish the global existence and asymptotic behavior of weak solutions for any α > 0 and γ > 1 under the assumption that the density function keeps a constant state at far fields. In particular, in the case that \documentclass[12pt]{minimal}\begin{document}$0<\alpha <\frac{1}{2}$\end{document}0<α<12, we obtain the large time behavior of the strong solution obtained by Mellet and Vasseur when the solution has a lower bound (no vacuum).

Lower bound of [formula omitted] decay of the Navier–Stokes flow in the half-space [formula omitted] and its asymptotic behavior in the frequency space

We consider the asymptotic behavior of weak solutions of the Navier–Stokes equations in the half-space R+n. We obtain the lower bound of the energy decay of the Navier–Stokes flow, by means of the profile of the initial data. Indeed, we construct a class of the initial data which causes the slow decay of the Navier–Stokes flow, with an explicit rate. Furthermore, we investigate the asymptotic behavior of concentration in the frequency space.

Asymptotic behavior of solutions of a two-phase problem of microwave heating in the one-dimensional case

The present paper is concerned with the asymptotic behavior of the system describing the material heating due to microwave radiation in the one-dimensional case. The material is assumed to have inhomogeneous structure and two-phase properties. The parabolic-hyperbolic system describing the heat in the one-dimensional case is studied. The results on the asymptotic behavior of weak solutions of the systems are obtained.