Asymptotic Behavior Of Global Solutions Research Articles

Rate of approach to the steady state for a diffusion-convection equation on annular domains

In this paper, we study the asymptotic behavior of global solutions of the equation ut = �u + e jr uj in the annulus Br,R, u(x, t) = 0 on @Br and u(x, t) = M � 0 on @BR. It is proved that there exists a constant Mc > 0 such that the problem admits a unique steady state if and only

Asymptotic behavior for a nematic liquid crystal model with different kinematic transport properties

We study the asymptotic behavior of global solutions to hydrodynamical systems modeling the nematic liquid crystal flows under kinematic transports for molecules of different shapes. The coupling system consists of Navier-Stokes equations and kinematic transport equations for the molecular orientations. We prove the convergence of global strong solutions to single steady states as time tends to infinity as well as estimates on the convergence rate both in 2D for arbitrary regular initial data and in 3D for certain particular cases.

Different asymptotic behavior of global solutions for a parabolic system with nonlinear gradient terms

In this paper we study the global existence and the different asymptotic behavior of mild solutions for the nonlinear parabolic system: ∂ t u = Δ u + a | ∇ v | p , ∂ t v = Δ v + b | ∇ u | q , t > 0 , x ∈ R N , where a , b ∈ R , N ⩾ 1 , 1 < p ⩽ q < 2 and p q > q N + 1 + N + 2 N + 1 . We prove, in particular, that if the initial values behave as u ( 0 , x ) ∼ ω 1 ( x / | x | ) | x | − α , v ( 0 , x ) ∼ ω 2 ( x / | x | ) | x | − β as | x | → ∞ , 0 < α , β < N , β + 2 − q q < α , α + 2 − p p < β and under suitable conditions on ω 1 , ω 2 , then the resulting solutions are global. Furthermore, although the scaling invariance properties of these initial values and the system are different, we prove that some of the solutions are asymptotic to self-similar solutions of appropriate asymptotic systems which depend on the values of α and β. The asymptotic behavior estimates are given in the W 1 , ∞ ( R N ) × W 1 , ∞ ( R N ) -norm and are stable under some small perturbations. The results of this paper complete those of Al-Elaiw and Tayachi (2010) [1] known only for β + 2 − q q = α and α + 2 − p p = β .

Existence, non-existence and asymptotic behavior of global solutions to the Cauchy problem for systems of semilinear hyperbolic equations with damping terms

We consider the Cauchy problem for systems of semilinear hyperbolic equations. Using the L p → L q type estimation for the corresponding linear parts, the existence and uniqueness of weak global solutions are investigated. We also established the behavior of solutions and their derivatives as t → + ∞ . Using the method of test functions developed in the works (Mitidieri and Pokhozhaev, 2001 [11], Veron and Pohozaev, 2001 [12] and Caristi, 2000 [23]) we obtain the analogue of the Fujita–Hayakawa type criterion for the absence of global solutions to some system of semilinear hyperbolic inequalities with damping. It follows that the conditions of existence theorem imposed on the growth of nonlinear parts are exact in some sense.

Global solutions to the Cauchy problem for a system of damped wave equations

The Cauchy problem to the system of nonlinear damped wave equations is treated. Several authors have shown existence and asymptotic behavior of global solutions to the above problem when the space dimension is not greater than three. We will show the existence and asymptotic behavior of global solutions to the problem with rapidly decaying initial data when the space dimension is greater than three, where we apply estimates in weighted Sobolev spaces of the above solution operator. Moreover, using the theory of modulation spaces introduced by Feitinger [4], we will also show the existence and asymptotic behavior of global solutions to the problem with slowly decaying initial data.

Asymptotic Symmetry for a Class of Quasi-Linear Parabolic Problems

Abstract We study the symmetry properties of the weak positive solutions to a class of quasi-linear elliptic problems having a variational structure. On this basis, the asymptotic behaviour of global solutions of the corresponding parabolic equations is also investigated. In particular, if the domain is a ball, the elements of the ω limit set are nonnegative radially symmetric solutions of the stationary problem.

Existence and Asymptotic Behavior of Global Solutions for a Class of Nonlinear Higher-Order Wave Equation

The initial boundary value problem for a class of nonlinear higher-order wave equation with damping and source term in a bounded domain is studied, where , is a nature number, and and are real numbers. The existence of global solutions for this problem is proved by constructing the stable sets and shows the asymptotic stability of the global solutions as time goes to infinity by applying the multiplier method.

Scattering of a 2D Schrödinger equation with exponential growth in the conformal space

AbstractWe investigate the asymptotic behavior of global solutions for nonlinear Schrödinger equation with a defocusing exponential nonlinearity in two space dimensions. We prove that any (global) solution approaches a unique free Schrödinger solution at the time infinity in the conformal space. Copyright © 2009 John Wiley &amp; Sons, Ltd.

Global solutions of singular parabolic equations arising from electrostatic MEMS

We study dynamic solutions of the singular parabolic problem ( P) { u t − Δ u = λ ∗ | x | α ( 1 − u ) 2 , ( x , t ) ∈ B × ( 0 , ∞ ) , u ( x , 0 ) = u 0 ( x ) ⩾ 0 , x ∈ B , u ( x , t ) = 0 , x ∈ ∂ B , where α ⩾ 0 and λ ∗ > 0 are two parameters, and B is the unit ball { x ∈ R N : | x | ⩽ 1 } with N ⩾ 2 . Our interest is focussed on ( P) with λ ∗ : = ( 2 + α ) ( 3 N + α − 4 ) 9 , for which ( P) admits a singular stationary solution in the form S ( x ) = 1 − | x | 2 + α 3 . This equation models dynamic deflection of a simple electrostatic Micro-Electro-Mechanical-System (MEMS) device. Under the assumption u 0 ≨︀ S ( x ) , we address the existence, uniqueness, regularity, stability or instability, and asymptotic behavior of weak solutions for ( P) . Given α ∗ ∗ : = 4 − 6 N + 3 6 ( N − 2 ) 4 , in particular we show that if either N ⩾ 8 and α > α ∗ ∗ or 2 ⩽ N ⩽ 7 , then the minimal compact stationary solution u λ ∗ of ( P) is stable and while S ( x ) is unstable. However, for N ⩾ 8 and 0 ⩽ α ⩽ α ∗ ∗ , ( P) has no compact minimal solution, and S ( x ) is an attractor from below not from above. Furthermore, the refined asymptotic behavior of global solutions for ( P) is also discussed for the case where N ⩾ 8 and 0 ⩽ α ⩽ α ∗ ∗ , which is given by a certain matching of different asymptotic developments in the large outer region closer to the boundary and the thin inner region near the singularity.

ASYMPTOTICALLY SELF-SIMILAR GLOBAL SOLUTIONS OF A DAMPED WAVE EQUATION

We study the existence and the asymptotic behavior of global solutions of the damped wave equation [Formula: see text] where a ∈ ℝ, α &gt;1, t &gt; 0, x ∈ ℝn, n = 1,2,3, with initial condition (u (0), ut (0)) = (φ,ψ). For α &gt; 2 and α &gt; 1+2 / n, we prove the existence of mild global solutions for small initial data with low regularity and which are not in L1(ℝn). Under the additional hypothesis, (2 &lt; α &lt; 5, when n = 3), we prove that some of these solutions are asymptotic to the self-similar solutions of the associated semi-linear heat equation [Formula: see text] with homogeneous slowly decreasing initial data behaving like c|x|-2 / (α-1) as |x|→ ∞.

CONVERGENCE TO EQUILIBRIUM FOR A PARABOLIC–HYPERBOLIC PHASE-FIELD SYSTEM WITH NEUMANN BOUNDARY CONDITIONS

This paper is concerned with the asymptotic behavior of global solutions to a parabolic–hyperbolic coupled system which describes the evolution of the relative temperature θ and the order parameter χ in a material subject to phase transitions. For the system with homogeneous Neumann boundary conditions for both ¸ and χ, under the assumption that the nonlinearities λ and ϕ are real analytic functions, we prove the convergence of a global solution to an equilibrium as time goes to infinity by means of a suitable Łojasiewicz–Simon type inequality.

Non-Newton Filtration Equation with Nonconstant Medium Void and Critical Sobolev Exponent

In this paper we consider the existence and asymptotic estimates of global solutions and finite time blowup of local solutions of quasilinear parabolic equation with critical Sobolev exponent and with lower energy initial value; we also describe the asymptotic behavior of global solutions with high energy initial value.

On bounded and unbounded global solutions of a supercritical semilinear heat equation

We consider the Cauchy problem ut = �u +| u| p−1 u, x ∈ R N ,t >0, u(x, 0) = u0(x), x ∈ R N , where u0 ∈ C0(R N ), the space of all continuous functions on R N that decay to zero at infinity, and p is supercritical in the sense that N ≥ 11 and p ≥ ((N − 2) 2 − 4N + 8 √ N − 1)/(N − 2)(N − 10). We first examine the domain of attraction of steady states (and also of general solutions) in a class of admissible functions. In particular, we give a sharp con- dition on the initial function u0 so that the solution of the above problem converges to a given steady state. Then we consider the asymptotic behavior of global solutions bounded above and below by classical steady states (such solutions have compact trajectories in C0(R N ), under the supremum norm). Our main result reveals an interesting possibility: the solution may approach a continuum of steady states, not settling down to any particular one of them. Finally, we prove the existence of global unbounded solutions, a phenomenon that does not occur for Sobolev- subcritical exponents.

Blow-Up and Convergence Results for a One-Dimensional Non-Local Parabolic Problem

Considering a one-dimensional non-local semilinear parabolic problem, it is shown that blow-up in finite time occurs for suitable large initial conditions. The asymptotic behavior of global solutions corresponding to small initial conditions is also investigated. Their convergence in $H^1$-norm to a well determinated stationary solution is proved.

Asymptotically self-similar global solutions of a semilinear parabolic equation with a nonlinear gradient term

We study the existence and the asymptotic behaviour of global solutions of the semilinear parabolic equation u(0) = ϧwhere a, b ∈ℝ, q &gt; 1, p &gt; 1. Forq=2p/(p+1) and ½ 1(p-1)&gt;1 (equivalently, q &gt; (n + 2)/(n + 1)), we prove the existence of mild global solutions for small initial data with respect to some norm. Some of those solutions are asymptotically self-similar.

Analytic semigroups and semilinear heat equations in hilbert spaces

The existence, uniqueness and asymptotic behavior of global solutions of semilinear heat equations in Hilbert spaces are studied by developing new results in the theory of analytic semigroups of bounded linear operators.

Asymptotic behavior of global solutions to the Navier–Stokes equations in $\mathbb R^3$

We construct global solutions to the NavierStokes equations with initial data small in a Besov space. Under additional assumptions we show that they behave asymptotically like self-similar solutions.

Asymptotic Behavior of Global Solutions for a Chemical Reaction Model

This paper deals with a system of reaction-diffusion equations modeling a chemical reaction process involving three substances. The mathematical system consists of three semilinear partial differential equations of parabolic type with homogenous Neumann boundary conditions. The goal is to determine whether each substance has a positive asymptotic limit by the method of upper and lower solutions.

Global solutions of semilinear heat equations in Hilbert spaces

The existence, uniqueness, regularity and asymptotic behavior of global solutions of semilinear heat equations in Hilbert spaces are studied by developing new results in the theory of one‐parameter strongly continuous semigroups of bounded linear operators. Applications to special semilinear heat equations in L 2(ℝn) governed by pseudo‐differential operators are given.

On the convergence properties of global solutions for some reaction-diffusion systems under Neumann boundary conditions

We are concerned with the asymptotic behavior of global solutions for a class of reaction-diffusion systems under homogeneous Neumann boundary conditions. An example of the system which we consider in this paper is what we call a diffusive epidemic model. After we show that every global solution uniformly converges to the corresponding constant function as $t \to \infty$, we investigate the rate of this convergence. We can obtain it with use of $L^p$-estimates, integral equations via analytic semigroups, fractional powers of operators and some imbedding relations.