Upper and lower bounds for the solutions of Markov renewal equations

Upper and lower bounds are studied for the solutions of Markov renewal equations. Some of their special cases are derived under specific marginal conditons and in an alternating environment. The method to construct the bounds is also explained in detail. At the end, these bounds are applied to a shock model and an age-dependent branching process under Markovian environment.     

Upper bounds of the rates of decay for solutions of the Boussinesq equations

In this paper,upper bounds of the L2-decay rate for the Boussinesq equations are considered.Using the L2 decay rate of solutions for the heat equation,and assuming that the solutions of the Boussinesq equations are smooth,we obtain the upper bounds of L2 decay rate for the smooth solutions and difference between the solutions of the Boussinesq equations and those of the heat system with the same initial data.The decay results may then be obtained by passing to the limit of approximating sequences of solutions.The main tool is the Fourier splitting method.     

Global existence and decay estimate of solutions to magneto-micropolar fluid equations

We are concerned with magneto-micropolar fluid equations (1.3)–(1.4). The global existence of solutions to the Cauchy problem is investigated under certain conditions. Precisely, for the magneto-micropolar-Navier–Stokes (MMNS) system, we obtain global existence and large time behavior of solutions near a constant states in ${\mathbb{R}}^3$. Appealing to a refined pure energy method, we first obtain a global existence theorem by assuming that the $H^3$ norm of the initial data is small, but the higher order derivatives can be arbitrary large. If the initial data belongs to homogeneous Sobolev norms ${\dot{H}}^{?s}$ $(0\le s<\frac{3}{2})$ or homogeneous Besov norms ${\dot{B}}_{2,\infty }^{?s}$ $(0<s\le \frac{3}{2})$, we obtain the optimal decay rates of the solutions and its higher order derivatives. As an immediate byproduct, we also obtain the usual $L^p?L^2$ $(1\le p\le 2)$ type of the decay rates without requiring that the $L^p$ norm of initial data is small. At last, we derive a weak solution to (1.3)–(1.4) in ${\mathbb{R}}^2$ with large initial data.     

On upper and lower bounds of higher order derivatives for solutions to the 2D micropolar fluid equations

The present paper is concerned with asymptotic behaviours of the solutions to the micropolar fluid motion equations in R². Upper and lower bounds are derived for the L² decay rates of higher order derivatives of solutions to the micropolar fluid flows. The findings are mainly based on the basic estimates of the linearized micropolar fluid motion equations and generalized Gronwall type argument.     

Global attractor and decay estimates of solutions to a class of nonlinear evolution equations

In this work, we prove the existence of global attractor for the nonlinear evolution equation u_tt?Δu?Δu_t?Δu_tt + g(x, u)=f(x) in X=(H²(Ω)∩H(Ω)) × (H²(Ω)∩H(Ω)). This improves a previous result of Xie and Zhong in (J. Math. Anal. Appl. 2007; 336 :54–69.) concerning the existence of global attractor in H(Ω) × H(Ω) for a similar equation. Further, the asymptotic behavior and the decay property of global solution are discussed. Copyright © 2010 John Wiley & Sons, Ltd.     

On the decay rate of solutions of linear parabolic equations for infinitely increasing time

The first boundary value problem with zero boundary values is considered for a one-dimensional linear parabolic equation. If the equation is sufficiently close to the heat equation, the rate of decreasing for the solution is connected with the number of zero level lines of the solution, nonvanishing for all values of time. Bibliography: 4 titles. Translated from Trudy Seminara imeni I.G. Petrovskogo, No. 17, pp. 118–127, 1994.     

Estimations on upper and lower bounds of solutions to a class of tensor complementarity problems

We introduce a class of structured tensors, called generalized row strictly diagonally dominant tensors, and discuss some relationships between it and several classes of structured tensors, including nonnegative tensors, Btensors, and strictly copositive tensors. In particular, we give estimations on upper and lower bounds of solutions to the tensor complementarity problem (TCP) when the involved tensor is a generalized row strictly diagonally dominant tensor with all positive diagonal entries. The main advantage of the results obtained in this paper is that both bounds we obtained depend only on the tensor and constant vector involved in the TCP;and hence, they are very easy to calculate.     

Upper and lower bounds for solutions of generalized two-point boundary value problems

This paper is concerned with two-point boundary value problems for systems of differential equations and integro-differential equations. If ?, ψ and Φ, Ψ are functions which satisfy certain differential (integro-differential) inequalities, then the given problem has a solutionu ^* such that ?≦u ^*≦ψ and Φ≦u ^*′≦Ψ.     

On a decay rate for 1D-viscous compressible barotropic fluid equations

The Navier-Stokes equations of a compressible barotropic fluid in 1D with zero velocity boundary conditions are considered. We study the case of large initial data in H ¹ as well as the mass force such that the stationary density is positive. The uniform lower bound for the density is proved. By constructing suitable Lyapunov functionals, decay rate estimates in L ²-norm and H ¹-norm are given. The decay rate is exponential if so the decay rate of the nonstationary part of the mass force is. The results are proved in the Eulerian coordinates for a wide class of increasing state functions including with any γ > 0 as well as functions of arbitrarily fast growth. We also extend the results for equations of a multicomponent compressible barotropic mixture (in the absence of chemical reactions). Received December 20, 2000; accepted February 27, 2001.     

Lower bounds for blow-up time in a class of nonlinear wave equations

This paper deals with the blow-up of solutions u(x, t) to a class of nonlinear hyperbolic problems. Under certain conditions on the data, we construct a lower bound for the blow-up time t* when blow-up occurs. A Sobolev-type inequality to be used in our investigation will also be established.     

Convergence and decay rate to equilibrium of bounded solutions of quasilinear parabolic equations

We study the convergence and decay rate to equilibrium of bounded solutions of the quasilinear parabolic equation

ut−diva(x,∇u)+f(x,u)=0     

The regularity of bidimensional solutions of the third grade fluids equations

It was recently proved in [1,2] that the third grade fluids equations have a unique global bidimensional solution provided that the initial velocity belongs to the Sobolev space H². Here, we complete this result by proving that this solution preserves the Sobolev regularity of the initial data, i.e., if the initial velocity belongs to H^m, m ≥ 2, then the evolved velocity v(t, ·) also belongs to H^m for every time t.     

Upper and lower bounds of temporal and spatial decays for the Navier-Stokes equations

We obtain the lower bounds of the temporal-spatial decays for weak solutions of the Navier-Stokes equations

     

Weak and smooth solutions to incompressible Navier-Stokes-Landau-Lifshitz-Maxwell equations

Considering the Navier-Stokes-Landau-Lifshitz-Maxwell equations, in dimensions two and three, we use Galerkin method to prove the existence of weak solution. Then combine the a priori estimates and induction technique, we obtain the existence of smooth solution.     

Global existence and time decay of small solutions to the landau-ginzburg type equations

We study the Cauchy problem for the nonlinear dissipative equations (0.1) uo∂u-αδu + Β|u|^2/n u = 0,x ∃ Rⁿ,t } 0,u(0,x) = u0(x),x ∃ Rⁿ, where α,Β ∃ C, ℜα 0. We are interested in the dissipative case ℜα 0, and ℜδ(α,Β)≥ 0, θ = |∫ u0(x)dx| ⊋ 0, where δ(α, Β) = ##|α|^n-1n^n/2 / ((n + 1)|α|² + α^{2 n/2}. Furthermore, we assume that the initial data u₀ ∃ L^p are such that (1 + |x|)^αu0 ∃ L¹, with sufficiently small norm ∃ = (1 + |x|)^α u₀ 1 + u₀ p, wherep 1, α ∃ (0,1). Then there exists a unique solution of the Cauchy problem (0.1)u(t, x) ∃ C ((0, ∞); L^∞) ∩ C ([0, ∞); L¹ ∩ L^p) satisfying the time decay estimates for allt0 u(t)||_∞ Cɛt^-n/2(1 + η log 〈t〉)^-n/2, if hg = θ^2/n 2_π ℜδ(α, Β) 0; u(t)||_∞ Cɛt^-n/2(1 + Μ log 〈t〉)^-n/4, if η = 0 and Μ = θ^4/n 4π)² (ℑδ(α, Β))² ℜ((1 + 1/n) υ1-1 υ2) 0; and u(t)||_∞ Cɛt^-n/2(1 + κ log 〈t〉)^-n/6, if η = 0, Μ = 0, κ 0, where υl,l = 1,2 are defined in (1.2), κ is a positive constant defined in (2.31).     

Nonoscillation and oscillation of solutions of a class of third order difference equations

In this paper, sufficient conditions are obtained for nonoscillation/oscillation of all solutions of a class of nonlinear third order difference equations of the form