Decay of Fourier modes of solutions to the dissipative surface quasi-geostrophic equations on a finite domain

We consider the two dimensional dissipative surface quasi-geostrophic equation on the unit square with mixed boundary conditions. Under some suitable assumptions on the initial stream function, we obtain existence and uniqueness of solutions in the form of a fast converging trigonometric series. We prove that the Fourier coefficients of solutions have a non-uniform decay: in one direction the decay is exponential and along the other direction it is only power like. We establish global wellposedness for arbitrary large initial data.     

The exponential decay of a system of nonlinear wave equations

The exponential decay of a system of nonlinear wave equations with initial boundary values is considered. We have some sufficient conditions that ensure that the energy admits exponential decay by a compactness uniqueness argument and the energy estimates.     

Fast propagation for KPP equations with slowly decaying initial conditions

This paper is devoted to the analysis of the large-time behavior of solutions of one-dimensional Fisher-KPP reaction-diffusion equations. The initial conditions are assumed to be globally front-like and to decay at infinity towards the unstable steady state more slowly than any exponentially decaying function. We prove that all level sets of the solutions move infinitely fast as time goes to infinity. The locations of the level sets are expressed in terms of the decay of the initial condition. Furthermore, the spatial profiles of the solutions become asymptotically uniformly flat at large time. This paper contains the first systematic study of the large-time behavior of solutions of KPP equations with slowly decaying initial conditions. Our results are in sharp contrast with the well-studied case of exponentially bounded initial conditions.     

General Energy Decay for a Viscoelastic Equation of Kirchhoff Type with Acoustic Boundary Conditions

This paper is concerned with a viscoelastic equation of Kirchhoff type with acoustic boundary conditions in a bounded domain of \(\mathbb {R}^{n}.\) We show that, under suitable conditions on the initial data, the solution exists globally in time. Then, we prove the general energy decay of global solutions by applying a lemma of Martinez, which allows us to get our decay result for a class of relaxation functions wider than that usually considered.     

ON GLOBAL EXISTENCE AND NONEXISTENCE FOR SOME NONLINEAR PARABOLIC EQUATIONS

0IntroductionInthispaper,weconsidertheinitial-boundaryvalueproblemforthefailliliarequationwherefiisaboundeddomaininR"withsmoothboulldaryoff,p22isacollstantalldlp(z,u)l5of'(tl" 'forsomea20andc>0.FOrp(x,ti)=Itll"'u,theauthorsofpaper[4,7llolwereillterestedillllollllegativesolutionandhadobtainedfollowillgresults(alsosee[12]).1)If25a 2相似文献   

Point‐wise decay estimate for the global classical solutions to quasilinear hyperbolic systems

In this paper, we first consider the Cauchy problem for quasilinear strictly hyperbolic systems with weak linear degeneracy. The existence of global classical solutions for small and decay initial data was established in (Commun. Partial Differential Equations 1994; 19 :1263–1317; Nonlinear Anal. 1997; 28 :1299–1322; Chin. Ann. Math. 2004; 25B :37–56). We give a new, very simple proof of this result and also give a sharp point‐wise decay estimate of the solution. Then, we consider the mixed initial‐boundary‐value problem for quasilinear hyperbolic systems with nonlinear boundary conditions in the first quadrant. Under the assumption that the positive eigenvalues are weakly linearly degenerate, the global existence of classical solution with small and decay initial and boundary data was established in (Discrete Continuous Dynamical Systems 2005; 12 (1):59–78; Zhou and Yang, in press). We also give a simple proof of this result as well as a sharp point‐wise decay estimate of the solution. Copyright © 2008 John Wiley & Sons, Ltd.     

Some remarks on Saint-Venant's principle

We consider a three-dimensional hyperelastic cylinder in Ω = D × [0, ∞]. We study the asymptotic behaviour of the deformations of the cross-sections in an equilibrium state. In this case we show that the solutions either have exponential decay or exponential growth. We give some initial conditions such that the latter case occurs.     

On the temporal decay of solutions to the two‐dimensional nematic liquid crystal flows

We consider the temporal decay estimates for weak solutions to the two‐dimensional nematic liquid crystal flows, and we show that the energy norm of a global weak solution has non‐uniform decay under suitable conditions on the initial data. We also show the exact rate of the decay (uniform decay) of the energy norm of the global weak solution.     

Three-dimensional initial boundary-value problem of the convection of a viscous weakly compressible fluid. II. Uniqueness and stability of generalized solutions

We continue our study of the three-dimensional initial boundary-value problem of the convection of a viscous thermally inhomogeneous weakly compressible fluid which fills a cavity in a solid body. We prove theorems on the uniqueness of the generalized solution of this problem and its continuity with respect to initial conditions and perturbations. We obtain estimates of exponential type for the decay of solutions (in the mean) for large time.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 9, pp. 1189–1202, September, 1994.     

Global well-posedness of coupled parabolic systems

The initial boundary value problem of a class of reaction-diffusion systems(coupled parabolic systems)with nonlinear coupled source terms is considered in order to classify the initial data for the global existence,finite time blowup and long time decay of the solution.The whole study is conducted by considering three cases according to initial energy:the low initial energy case,critical initial energy case and high initial energy case.For the low initial energy case and critical initial energy case the sufficient initial conditions of global existence,long time decay and finite time blowup are given to show a sharp-like condition.In addition,for the high initial energy case the possibility of both global existence and finite time blowup is proved first,and then some sufficient initial conditions of finite time blowup and global existence are obtained,respectively.     

具有非线性梯度吸收项的快速扩散方程的有限熄灭

本文研究快速扩散方程ut-Δum +｜ u｜p =0的柯西问题 ,其中m ,p∈ ( 0 ,1) .对于 0

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Remarks on asymptotic behaviors of strong solutions to a viscous Boussinesq system

In this paper, we first address the space‐time decay properties for higher‐order derivatives of strong solutions to the Boussinesq system in the usual Sobolev space. The decay rates obtained here are optimal. The proof is based on a parabolic interpolation inequality, bootstrap argument, and some weighted estimates. Secondly, we present a new solution integration formula for the Boussinesq system, which will be employed to establish the existence of strong solutions for small initial data in some scaling invariant function spaces. The smallness conditions are somehow weaker than those presented by Brandolese and Schonbek. We further investigate the asymptotic profiles and decay properties of these strong solutions. Copyright © 2016 John Wiley & Sons, Ltd.     

Linear behaviour of solutions of a superlinear heat equation

We give sufficient conditions which guarantee that solutions of a superlinear heat equation decay to zero at the same rate as the solutions of the linear heat equation with the same initial data. This improves a previous result of Lee and Ni by showing that this behaviour holds for a significantly larger, and rather tightly defined class of solutions.     

Local solvability and blowup of the solution of the Rosenau-Bürgers equation with different boundary conditions

We consider several models of initial boundary-value problems for the Rosenau-Bürgers equation with different boundary conditions. For each of the problems, we prove the unique local solvability in the classical sense, obtain a sufficient condition for the blowup regime, and estimate the time of the solution decay. The proof is based on the well-known test-function method.     

Asymptotic behavior of solutions to a class of nonlinear wave equations of sixth order with damping

We investigate the initial value problem for a class of nonlinear wave equations of sixth order with damping. The decay structure of this equation is of the regularity‐loss type, which causes difficulty in high‐frequency region. By using the Fourier splitting frequency technique and energy method in Fourier space, we establish asymptotic profiles of solutions to the linear equation that is given by the convolution of the fundamental solutions of heat and free wave equation. Moreover, the asymptotic profile of solutions shows the decay estimate of solutions to the corresponding linear equation obtained in this paper that is optimal under some conditions. Finally, global existence and optimal decay estimate of solutions to this equation are also established. Copyright © 2016 John Wiley & Sons, Ltd.     

Numerical solution of the incompressible Navier-Stokes equations with explicit integration of the pressure term

We propose a formulation of the incompressible Navier-Stokes equations considering a Poisson equation with Neumann boundary conditions for the pressure, and innovative boundary conditions for the velocity. Numerical tests show that the proposed formulation ensures solenoidality of the velocity field. If the initial condition is not divergence-free, exponential decay is observed in time for the error in the fulfillment of the continuity equation. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

具有内部点耗散的Timoshenko梁的能量衰减估计

研究具有反馈控制力的Timoshenko梁的能量衰减．证明了梁的能量不是一致衰减的．当梁的能量不是一致衰减时,利用初始值的正则性和无阻尼问题的最佳正则性结果,给出了多项式衰减估计．     

Uniform stability of semilinear wave equations with arbitrary local memory effects versus frictional dampings

This paper is concerned with the mixed initial–boundary value problem for semilinear wave equations with complementary frictional dampings and memory effects. We successfully establish uniform exponential and polynomial decay rates for the solutions to this initial–boundary value problem under much weak conditions concerning memory effects. More specifically, we obtain the exponential and polynomial decay rates after removing the fundamental condition that the memory-effect region includes a part of the system boundary, while the condition is a necessity in the previous literature; moreover, for the polynomial decay rates we only assume minimal conditions on the memory kernel function g, without the usual assumption of $g^{\prime }$ controlled by g.     

Optimal time decay of the quantum Landau equation in the whole space

We are concerned with the Cauchy problem of the quantum Landau equation in the whole space. The existence of local in time nearby quantum Maxwellian solutions is proved by the iteration method and generalized maximum principle. Based on Kawashima?s compensating function and nonlinear energy estimates, the global existence and the optimal time decay rate of those solutions are obtained under some conditions on initial data.     

Blow-up phenomena and global existence for the nonuniform weakly dissipative b-equation

We first establish the local well-posedness for the nonuniform weakly dissipative b-equation which includes both the weakly dissipative Camassa-Holm equation and the weakly dissipative Degasperis-Procesi equation as its special cases. We then study the blow-up phenomena and the long time behavior of the solutions. Two blow-up results are established for certain initial profiles. Moreover, two sufficient conditions for the decay of the solutions are presented.