Long-time Behavior Of Solutions Research Articles

Long time behavior of solutions of an electroconvection model in $${\mathbb {R}}^2$$

Long time behavior of solutions of an electroconvection model in $${\mathbb {R}}^2$$

Global well posedness for the semilinear edge-degenerate parabolic equations on singular manifolds

Abstract In this article, we study the long-time dynamical behavior of the solution for a class of semilinear edge-degenerate parabolic equations on manifolds with edge singularities. By introducing a family of potential well and compactness method, we reveal the dependence between the initial data and the long-time dynamical behavior of the solution. Specifically, we give the threshold condition for the initial data, which makes the solution exist globally or blowup in finite-time with subcritical, critical, and supercritical initial energy, respectively. Moreover, we also discussed the long-time behavior of the global solution, the estimate of blowup time, and blowup rate. Our results show that the relationship between the initial data and the long-time behavior of the solution can be revealed in the weighted Sobolev spaces for nonlinear parabolic equations on manifolds with edge singularities.

Nonlinear enhanced dissipation in viscous Burgers type equations II

In this follow up but self contained paper, we focus on the viscous Burgers equation. There, using the Hopf-Cole transformation, we compute the long time behavior of solutions for some classes of infinite mass initial datas. We show that an enhanced dissipation effect occurs generically, that is the decay rate in time is better than if we considered instead the heat equations for the same inital value. We also show the existence of a kind of global attractor per class.

Long time behavior of solutions for time-fractional pseudo-parabolic equations involving time-varying delays and superlinear nonlinearities

Long time behavior of solutions for time-fractional pseudo-parabolic equations involving time-varying delays and superlinear nonlinearities

Sticky nonlinear SDEs and convergence of McKean–Vlasov equations without confinement

AbstractWe develop a new approach to study the long time behaviour of solutions to nonlinear stochastic differential equations in the sense of McKean, as well as propagation of chaos for the corresponding mean-field particle system approximations. Our approach is based on a sticky coupling between two solutions to the equation. We show that the distance process between the two copies is dominated by a solution to a one-dimensional nonlinear stochastic differential equation with a sticky boundary at zero. This new class of equations is then analyzed carefully. In particular, we show that the dominating equation has a phase transition. In the regime where the Dirac measure at zero is the only invariant probability measure, we prove exponential convergence to equilibrium both for the one-dimensional equation, and for the original nonlinear SDE. Similarly, propagation of chaos is shown by a componentwise sticky coupling and comparison with a system of one dimensional nonlinear SDEs with sticky boundaries at zero. The approach applies to equations without confinement potential and to interaction terms that are not of gradient type.

Numerical study of fractional Camassa–Holm equations

A numerical study of fractional Camassa–Holm equations is presented. Smooth solitary waves are constructed numerically. Their stability is studied as well as the long time behavior of solutions for general localized initial data from the Schwartz class of rapidly decreasing functions. The appearance of dispersive shock waves is explored.

Non–uniform decay of solutions to the incompressible magneto–micropolar fluids with/without magnetic diffusion and spin viscosity

We are concerned with the long time behavior of solutions to the incompressible magneto–micropolar fluids with/without magnetic diffusion and spin viscosity in two and three dimensions. When the magnetic diffusion is absent, we show that L2–norm of fluid velocity and micro–rotational velocity vanishes and L2–norm of the magnetic field converges to a non–negative constant as time tends to infinity. Moreover, when both magnetic diffusion and spin viscosity are present and the initial data belongs to L2, we prove that the solutions decay to zero without a rate, and this non–uniform decay is optimal. The proofs are based on Fourier splitting method, low–frequency and high–frequency decomposition techniques and delicate energy estimates.

Dynamic behavior of reaction–diffusion equation with Holling II predation

In this paper, we consider the positive solutions for some reaction–diffusion equations with Holling II predation. The main aim is to analyze the long time behavior of solutions, both the diffusion and non-diffusion problems are investigated. Our results also exhibit different effects between Dirichlet and Neumann boundary conditions.

Dispersive estimates for the wave equation inside cylindrical convex domains

The dispersive and Strichartz estimates are essential for establishing well posedness results for nonlinear equations as well as long time behaviour of solutions to the equation. While in the boundary-less case these estimates are well understood, the case of boundary the situation can become much more difficult. In this work, we establish local in time dispersive estimates for solutions of the model case Dirichlet wave equation inside cylindrical convex domains \(\Omega\subset\mathbb{R}^3\) with smooth boundary \(\partial\Omega\neq \emptyset\). In this paper, we provide detailed proofs of the results established in [16, 17]. Let us recall that dispersive estimates are key ingredients to prove Strichartz estimates. Strichartz estimates for waves inside an arbitrary domain \(\Omega\) have been proved by Blair, Smith, Sogge [4, 5]. Optimal estimates in strictly convex domains have been obtained in [12]. Our case of cylindrical domains is an extension of the result of [12] in the case when the nonnegative curvature radius depends on the incident angle and vanishes in some directions.

Global Existence and Blow-up Solutions for a Parabolic Equation with Critical Nonlocal Interactions

AbstractIn this paper, we study the initial boundary value problem for the nonlocal parabolic equation with the Hardy–Littlewood–Sobolev critical exponent on a bounded domain. We are concerned with the long time behaviors of solutions when the initial energy is low, critical or high. More precisely, by using the modified potential well method, we obtain global existence and blow-up of solutions when the initial energy is low or critical, and it is proved that the global solutions are classical. Moreover, we obtain an upper bound of blow-up time for $$J_{\mu }(u_{0})<0$$ J μ ( u 0 ) < 0 and decay rate of $$H^{1}_{0}$$ H 0 1 and $$L^{2}$$ L 2 -norm of the global solutions. When the initial energy is high, we derive some sufficient conditions for global existence and blow-up of solutions. In addition, we are going to consider the asymptotic behavior of global solutions, which is similar to the Palais-Smale (PS for short) sequence of stationary equation.

Dynamics and regularity for non-autonomous reaction-diffusion equations with anomalous diffusion

This paper is concerned with the long time behavior of solutions for a non-autonomous reaction-diffusion equations with anomalous diffusion. Under suitable assumptions on nonlinearity and external force, the global well-posedness has been studied. Then the pullback attractors in L 2 ( Ω ) and H 0 α ( Ω ) ( 0 < α < 1) have been achieved with a restriction on the growth order of nonlinearity as 2 ⩽ p ⩽ 2 ( n − α ) n − 2 α . The results presented can be seen as the extension for classical theory of infinite dimensional dynamical system to the fractional diffusion equations.

The effect of Lévy noise and white noise on a Leslie-Gower predator-prey system with prey refuge

A stochastic Leslie-Gower predator-prey system with prey refuge is discussed, L$ \acute{e} $vy noise and white noise are adopted to simulate the random change of environment. The existence of global positive solution and the stochastic boundedness is proved, the threshold conditions of extinction and stable in the mean are given, the long time behavior of solution for the predator-prey system is also considered. In the end, some examples of numerical simulation are given to verify these conclusions. Compared to the deterministic system, the prey population and the predator population will be extinct, when the intensity of noise is large enough, however, it does not happen in the deterministic system, where the positive equilibrium is always globally stable. These results indicate that the stochastic change of environment has great influence on the population dynamic system.

Long-time behaviour for a nonlocal model from directed polymers

We consider the long time behaviour of solutions to a nonlocal reaction diffusion equation that arises in the study of directed polymers in a random environment. The model is characterized by convolution with a kernel R and an L 2 inner product. In one spatial dimension, we extend a previous result of the authors (arXiv:2002.02799), where only the case was considered; in particular, we show that solutions spread according to a power law consistent with the KPZ scaling conjectured for directed polymers. In the special case when , we find the exact profile of the solution in the rescaled coordinates. We also consider the behaviour in higher dimensions. When the dimension is three or larger, we show that the long-time behaviour is the same as the heat equation in the sense that the solution converges to a standard Gaussian. In contrast, when the dimension is two, we construct a non-Gaussian self-similar solution.

Decay Rate Estimates for the Wave Equation with Subcritical Semilinearities and Locally Distributed Nonlinear Dissipation

We study the stabilization and the wellposedness of solutions of the wave equation with subcritical semilinearities and locally distributed nonlinear dissipation. The novelty of this paper is that we deal with the difficulty that the main equation does not have good nonlinear structure amenable to a direct proof of a priori bounds and a desirable observability inequality. It is well known that observability inequalities play a critical role in characterizing the long time behaviour of solutions of evolution equations, which is the main goal of this study. In order to address this, we truncate the nonlinearities, and thereby construct approximate solutions for which it is possible to obtain a priori bounds and prove the essential observability inequality. The treatment of these approximate solutions is still a challenging task and requires the use of Strichartz estimates and some microlocal analysis tools such as microlocal defect measures. We include an appendix on the latter topic here to make the article self contained and supplement details to proofs of some of the theorems which can be already be found in the lecture notes of Burq and Gérard (http://www.math.u-psud.fr/~burq/articles/coursX.pdf, 2001). Once we establish essential observability properties for the approximate solutions, it is not difficult to prove that the solution of the original problem also possesses a similar feature via a delicate passage to limit. In the last part of the paper, we establish various decay rate estimates for different growth conditions on the nonlinear dissipative effect. We in particular generalize the known results on the subject to a considerably larger class of dissipative effects.

Study of Weak Solutions for a Class of Degenerate Parabolic Variational Inequalities with Variable Exponent

The authors of this text study weak solutions to the variational inequalities with degenerate parabolic operators and symmetric structure in Sobolev spaces with variable exponent. The existence and uniqueness of the solutions are treated by using the penalty method and the reduction method in the weak sense. The authors also discuss the nonexistence and long-time behavior of solutions.

Local zero-stability of rough evolution equations

We analyze the long time behavior of solutions to rough parabolic equations. More precisely, we show local exponential stability for the mild solution driven by a fractional Brownian motion with Hurst parameter [Formula: see text].

Global existence and finite time blow‐up for the heat flow of H‐system with constant mean curvature

In this paper, we use the modified potential well method to study the long time behaviors of solutions to the heat flow of H‐system in a bounded smooth domain of . Global existence and finite time blowup of solutions are proved when the initial energy is in three cases. When the initial energy is low or critical, we not only give a threshold result for the global existence and blowup of solutions but also obtain the decay rate of the norm for global solutions. When the initial energy is high, sufficient conditions for the global existence and blowup of solutions are also provided. We extend the recent results which were.

Long-time behavior of solutions to free boundary problems modeling tumor evolution

In this paper, we consider some free boundary problems that arise in the mathematical modeling of cancer and its treatment. In these models, a parameter η is introduced to represent either the killing rate of the cancer cells by the T cells or a drug called checkpoint inhibitor. We study the long-time behavior of the tumor as the parameter η varies.

Absence of singularities in solutions for the compressible Euler equations with source terms in

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.

Global existence and blow-up for semilinear parabolic equation with critical exponent in R N

In this paper, we use the self-similar transformation and the modified potential well method to study the long time behaviors of solutions to the classical semilinear parabolic equation associated with critical Sobolev exponent in R N . Global existence and finite time blowup of solutions are proved when the initial energy is in three cases. When the initial energy is low or critical, we not only give a threshold result for the global existence and blowup of solutions, but also obtain the decay rate of the L 2 norm for global solutions. When the initial energy is high, sufficient conditions for the global existence and blowup of solutions are also provided. We extend the recent results which were obtained in [R. Ikehata, M. Ishiwata, T. Suzuki, Ann. Inst. H. Poincaré Anal. Non Linéaire 27(2010), No. 3, 877–900].