Time Behavior Of Solutions Research Articles

On the behavior in time of solutions to motion of Non-Newtonian fluids

We study the behavior on time of weak solutions to the non-stationary motion of an incompressible fluid with shear rate dependent viscosity in bounded domains when the initial velocity {u}_0 {in } {L}^2. Our estimates show the different behavior of the solution as the growth condition of the stress tensor varies. In the “dilatant” or “shear thickening” case we prove that the decay rate does not depend on u_0, then our estimates also apply for irregular initial velocity.

Critical Fujita Exponent for the Porous Medium Equation in ${\mathbb R}^{{N}}$ with hole

We study the large time behavior of solutions to the Neumann Problem of the porous medium equation. It is shown that the critical Fujita exponent is determined not only by the spatial dimension and the nonlinear exponent, but also by the coefficient $k$ of the first-order term.

Dirichlet-boundary value problem for one dimensional nonlinear Schrödinger equations with large initial and boundary data

We consider the inhomogeneous Dirichlet-boundary value problem with large initial and boundary data for nonlinear Schrodinger equations in one space dimension. Global existence and asymptotic behavior in time of solutions to the problem are obtained by using the classical energy method and factorization techniques.

Free boundary problems with nonlocal and local diffusions I: Global solution

We study a class of free boundary problems of ecological models with nonlocal and local diffusions, which are natural extensions of free boundary problems of reaction diffusion systems in there local diffusions are used to describe the population dispersal, with the free boundary representing the spreading front of the species. We prove that such kind of nonlocal and local diffusion problems has a unique global solution, and then show that a spreading-vanishing dichotomy holds. Moreover, criteria of spreading and vanishing, and long time behavior of solution when spreading happens are established for the classical Lotka-Volterra competition and prey-predator models. Compared with free boundary problems of reaction diffusion systems with local diffusions ([10,24,25]), with nonlocal diffusions ([9]) as well as with nonlocal and local diffusions ([14]) (one equation is Cauchy problem and the other one is free boundary problem), the present paper involves some new difficulties, which should be overcome by use of new techniques. This is part I of a two part series, where we prove the existence, uniqueness, regularity and estimates of global solution. The spreading-vanishing dichotomy, criteria of spreading and vanishing, and long-time behavior of solution when spreading happens will be studied in the separate part II ([17]).

Free boundary problems with nonlocal and local diffusions Ⅱ: Spreading-vanishing and long-time behavior

<p style='text-indent:20px;'>This is part Ⅱ of our study on the free boundary problems with nonlocal and local diffusions. In part Ⅰ, we obtained the existence, uniqueness, regularity and estimates of global solution. In part Ⅱ here, we show a spreading-vanishing dichotomy, and provide the criteria of spreading and vanishing, as well as the long time behavior of solution when spreading happens.

A time-periodic reaction–diffusion–advection equation with a free boundary and sign-changing coefficients

We consider a reaction–diffusion–advection equation of the form: ut=uxx−β(t)ux+f(t,u) for x∈[0,h(t)), where β(t) is a T-periodic function, f(t,u) is a T-periodic Fisher–KPP type of nonlinearity with a(t)≔fu(t,0) changing sign, h(t) is a free boundary satisfying the Stefan condition. We study the long time behavior of solutions and find that there are two critical numbers c̄ and B(β̃) with B(β̃)>c̄>0, β̄≔1T∫0Tβ(t)dt and β̃(t)≔β(t)−β̄, such that a vanishing–spreading dichotomy result holds when |β̄|<c̄; a vanishing–transition–virtual spreading trichotomy result holds when β̄∈[c̄,B(β̃)); all solutions vanish when β̄⩾B(β̃) or β̄⩽−c̄.

Asymptotically self-similar behaviour of global solutions for semilinear heat equations with algebraically decaying initial data

AbstractWe consider the Cauchy problem $$\left\{ {\matrix{ {u_t = \Delta u + u^p,\quad } \hfill &amp; {x\in {\bf R}^N,\;t \leq 0,} \hfill \cr {u(x,0) = u_0(x),\quad } \hfill &amp; {x\in {\bf R}^N,} \hfill \cr } } \right.$$where N &gt; 2, p &gt; 1, and u0 is a bounded continuous non-negative function in RN. We study the case where u0(x) decays at the rate |x|−2/(p−1) as |x| → ∞, and investigate the convergence property of the global solutions to the forward self-similar solutions. We first give the precise description of the relationship between the spatial decay of initial data and the large time behaviour of solutions, and then we show the existence of solutions with a time decay rate slower than the one of self-similar solutions. We also show the existence of solutions that behave in a complicated manner.

Asymptotic behavior in time of solution to the nonlinear Schrödinger equation with higher order anisotropic dispersion

We consider the asymptotic behavior in time of solutions to the nonlinear Schrodinger equation with fourth order anisotropic dispersion (4NLS) which describes the propagation of ultrashort laser pulses in a medium with anomalous time-dispersion in the presence of fourth-order time-dispersion. We prove existence of a solution to (4NLS) which scatters to a solution of the linearized equation of (4NLS) as \begin{document}$t\to∞$\end{document} .

Asymptotic Stability of Combination of Contact Discontinuity with Rarefaction Waves for the One-Dimensional Viscous Micropolar Fluid Model

In this paper, we consider with the large time behavior of solutions of the Cauchy problem to the one-dimensional compressible micropolar fluid model, where the far field states are prescribed. When the corresponding Riemann problem for the Euler system admits the solution consisting of contact discontinuity and rarefaction waves, it is shown that the combination wave corresponding to the contact discontinuity, with rarefaction waves is asymptotically stable provided that the strength of the combination wave and the initial perturbation are suitably small. This result is proved by using elementary L2-energy methods.

Asymptotic Time-Behaviour of Solutions to Scalar Conservation Law with a Convex Flux Function

We consider the long-time behaviour of solutions of the Cauchy problem for a quasilinear equation ut + f(u)x = 0 with a strictly convex flux function f(u) and initial function u0(x) having the the one-sided limiting mean values u± that are uniform with respect to translations. The estimates of the rates of convergence to solutions of the Riemann problem depending on the behaviour of the integrals $ \int\limits_a^{a + y} {\left( {{u_0}\left( x \right) - {u^ \pm }} \right)} dx $ as y→±∞ are established. The similar results are obtained for solutions of the mixed problem in the domain x &gt; 0, t &gt; 0 with a constant boundary data u– and initial data having limiting mean value u±.

Global existence and large time behavior of solutions for full compressible Hall-MHD equations

ABSTRACT In this paper, we first provide a derivation of compressible nonisentropic Hall-magnetohydrodynamic (Hall-MHD) system from generalized Ohm's law. Then, we prove the global existence of strong solutions for the compressible nonisentropic Hall-MHD model based on the standard energy method in -framework. Finally, optimal decay rates of strong solutions in -norm are obtained if the initial data belong to additionally.

A two-species weak competition system of reaction-diffusion-advection with double free boundaries

<p style='text-indent:20px;'>In this paper, we investigate a two-species weak competition system of reaction-diffusion-advection with double free boundaries that represent the expanding front in a one-dimensional habitat, where a combination of random movement and advection is adopted by two competing species. The main goal is to understand the effect of small advection environment and dynamics of the two species through double free boundaries. We provide a spreading-vanishing dichotomy, which means that both of the two species either spread to the entire space successfully and survive in the new environment as time goes to infinity, or vanish and become extinct in the long run. Furthermore, if the spreading or vanishing of the two species occurs, some sufficient conditions via the initial data are established. When spreading of the two species happens, the long time behavior of solutions and estimates of spreading speed of both free boundaries are obtained.

Large Time Behavior of Solutions to Parabolic Equations with Dirichlet Operators and Nonlinear Dependence on Measure Data

We study large time behavior of renormalized solutions of the Cauchy problem for equations of the form ∂tu − Lu + λu = f(x, u) + g(x, u) ⋅ μ, where L is the operator associated with a regular lower bounded semi-Dirichlet form and μ is a nonnegative bounded smooth measure with respect to the capacity determined by . We show that under the monotonicity and some integrability assumptions on f, g as well as some assumptions on the form , u(t, x) → v(x) as t →∞ for quasi-every x, where v is a solution of some elliptic equation associated with our parabolic equation. We also provide the rate convergence. Some examples illustrating the utility of our general results are given.

Large-Time Behavior of an Infinite System of Harmonic Oscillators on the Half-Line

An initial–boundary value problem for an infinite one-dimensional chain of harmonic oscillators on the half-line is considered. The large time behavior of solutions is studied and dispersive bounds are derived.

Self-similar point vortices and confinement of vorticity

ABSTRACTThis papers deals with the large time behavior of solutions of the incompressible Euler equations in dimension 2. We consider a self-similar configuration of point vortices which grows like the square root of the time. We study the confinement properties of a blob of vorticity initially located around the first point vortex and moving in the velocity field produced by itself and by the other point vortices. We find a sufficient condition on the point vortices such that the vorticity stays confined around the first point vortex at a rate better than the square root of the time. The relevance to the large time behavior of the Euler equations is discussed.

Analysis of a System Describing Proliferative-Quiescent Cell Dynamics

Systems describing the dynamics of proliferative and quiescent cells are commonly used as computational models, for instance for tumor growth and hematopoiesis. Focusing on the very earliest stages of hematopoiesis, stem cells and early progenitors, the authors introduce a new method, based on an energy/Lyapunov functional to analyze the long time behavior of solutions. Compared to existing works, the method in this paper has the advantage that it can be extended to more complex situations. The authors treat a system with space variable and diffusion, and then adapt the energy functional to models with three equations.

Complex Ginzburg–Landau equations with dynamic boundary conditions

The initial-dynamic boundary value problem (idbvp) for the complex Ginzburg–Landau equation (CGLE) on bounded domains of RN is studied by converting the given mathematical model into a Wentzell initial–boundary value problem (ibvp). First, the corresponding linear homogeneous idbvp is considered. Secondly, the forced linear idbvp with both interior and boundary forcings is studied. Then, the nonlinear idbvp with Lipschitz nonlinearity in the interior and monotone nonlinearity on the boundary is analyzed. The local well-posedness of the idbvp for the CGLE with power type nonlinearities is obtained via a contraction mapping argument. Global well-posedness for strong solutions is shown. Global existence and uniqueness of weak solutions are proven. Smoothing effect of the corresponding evolution operator is proved. This helps to get better well-posedness results than the known results on idbvp for nonlinear Schrödinger equations (NLS). An interesting result of this paper is proving that solutions of NLS subject to dynamic boundary conditions can be obtained as inviscid limits of the solutions of the CGLE subject to same type of boundary conditions. Finally, long time behavior of solutions is characterized and exponential decay rates are obtained at the energy level by using control theoretic tools.

A reaction–diffusion–advection competition model with two free boundaries in heterogeneous time-periodic environment

In this paper, we study the dynamics of a two-species competition model with two different free boundaries in heterogeneous time-periodic environment, where the two species adopt a combination of random movement and advection upward or downward along the resource gradient. We show that the dynamics of this model can be classified into four cases, which forms a spreading-vanishing quartering. The notion of the minimal habitat size for spreading is introduced to determine if species can always spread. Rough estimates of the asymptotic spreading speed of free boundaries and the long time behavior of solutions are also established when spreading occurs. Furthermore, some sufficient conditions for spreading and vanishing are provided.

The global attractors and their Hausdorff and fractal dimensions estimation for the higher-order nonlinear Kirchhoff-type equation with nonlinear strongly damped terms

In this paper ,we study the long time behavior of solution to the initial boundary value problems for higher -orderkirchhoff-type equation with nonlinear strongly dissipation:At first ,we prove the existence and uniqueness of the solution by priori estimate and Galerkin methodthen we establish the existence of global attractors ,at last,we consider that estimation of upper bounds of Hausdorff and fractal dimensions for the global attractors are obtain.

Pattern Formations for a Strong Interacting Free Boundary Problem

In this paper we consider the system of reaction-diffusion-advection equations with a free boundary, which arises in a competition ecological model in heterogeneous environment. In strong competition case, we study the influence of competition rates on the long time behavior of solutions and prove that two species spatially segregate as the competition rates become large. Besides, by using a blow up method, we obtain the uniform Holder bounds for solutions of the system.