Blow-Up and Decay Estimate in a Logarithmic p-Laplace Parabolic Equation

This paper deals with an initial-boundary value problem of a fourth-order parabolic equation involving Logarithmic type p-Laplacian,which could be proposed as a model for the epitaxial growth of thin films.By using the variational method and the logarithmic type Sobolev inequality,we give some threshold results for blow-up solutions and global solutions,which could be classified by the initial energy.The asymptotic estimates about blow-up time and decay estimate of weak solutions are obtained.

DECAY ESTIMATES FOR ISENTROPIC COMPRESSIBLE MAGNETOHYDRODYNAMIC EQUATIONS IN BOUNDED DOMAIN

In this paper, under the hypothesis that y is upper bounded, we construct a Lyapunov functional for the multidimensional isentropic compressible magnetohydrodynamic equations and show that the weak solutions decay exponentially to the equilibrium state in L2 norm. Our result verifies that the method of Daoyuan Fang, Ruizhao Zi and Ting Zhang I1] can be adapted to magnetohydrodynamic equations.

L^p-L^q decay estimates of solutions to Cauchy problems of thermoviscoelastic systems

L^p- L^q decay estimate of solution to Cauchy problem of a linear thermoviscoelastic system is studied. By using a diagonalization argument of frequency analysis, the coupled system will be decoupled micrologically. Then with the help of the information of characteristic roots for the coefficient matrix of the system, L^p- L^q decay estimate of parabolic type of solution to the Cauchy problem is obtained.

Spatial Decay Estimates for the Solutions to Stokes Equations in Four Kinds of Semi-Infinite Cylinders

This paper investigates the spatial behavior of the solutions of the Stokes equations in a semi-infinite cylinder.We consider four kinds of semi-infinite cylinders with boundary conditions of Dirichlet type.For each type of cylinder we obtain the spatial decay estimates for the solutions.To make the attenuation meaningful,we derive the explicit bound for the total energy in terms of the initial boundary data.

ON THE DECAY AND SCATTERING FOR THE KLEIN-GORDON-HARTREE EQUATION WITH RADIAL DATA

In this paper,we study the decay estimate and scattering theory for the Klein-Gordon-Hartree equation with radial data in space dimension d≥3.By means of a compactness strategy and two Morawetz-type estimates which come from the linear and nonlinear parts of the equation,respectively,we obtain the corresponding theory for energy subcritical and critical cases.The exponent range of the decay estimates is extended to 0〈γ≤4 and γ〈d with Hartree potential V（x） =｜x｜-γ.

STRESS EFFECT DECAY ESTIMATES FOR ANISOTROPIC MATERIAL IN A SEMI-INFINITE STRIP

In this paper, Saint-Venant's principle for anisotropic material in an end-loaded. semi-infinite elastic strip is established. Energy method is used to establish the lower bounds of the decay estimate of stress effect. An explicit estimate formula in terms of the elastic constants of the anisotropic materials is presented. Finally, a numerical example for an end-loaded, off-axis, graphite-epoxy strip is given to illustrate the results.

SHARP MORREY REGULARITY THEORY FOR A FOURTH ORDER GEOMETRICAL EQUATION

This paper is a continuation of recent work by Guo-Xiang-Zheng[10].We deduce the sharp Morrey regularity theory for weak solutions to the fourth order nonhomogeneous Lamm-Rivière equation △^{2}u=△(V▽u)+div(w▽u)+(▽ω+F)·▽u+f in B^(4),under the smallest regularity assumptions of V,ω,ω,F,where f belongs to some Morrey spaces.This work was motivated by many geometrical problems such as the flow of biharmonic mappings.Our results deepens the Lp type regularity theory of[10],and generalizes the work of Du,Kang and Wang[4]on a second order problem to our fourth order problems.

GLOBAL CLASSICAL SOLUTIONS OF SEMILINEAR WAVE EQUATIONS ON R^(3)×T WITH CUBIC NONLINEARITIES

In this paper,we establish global classical solutions of semilinear wave equations with small compact supported initial data posed on the product space R^(3)×T.The semilinear nonlinearity is assumed to be of the cubic form.The main ingredient here is the establishment of the L^(2)-L^(∞)decay estimates and the energy estimates for the linear problem,which are adapted to the wave equation on the product space.The proof is based on the Fourier mode decomposition of the solution with respect to the periodic direction,the scaling technique,and the combination of the decay estimates and the energy estimates.

The Exact Limits and Improved Decay Estimates for All Order Derivatives of the Global Weak Solutions of n-Dimensional Incompressible Navier-Stokes Equations

We couple together existing ideas,existing results,special structure and novel ideas to accomplish the exact limits and improved decay estimates with sharp rates for all order derivatives of the global weak solutions of the Cauchy problem for an n-dimensional incompressible Navier-Stokes equations.We also use the global smooth solution of the corresponding heat equation to approximate the global weak solutions of the incompressible Navier-Stokes equations.

Weak solution for a fourth-order nonlinear wave equation

The existence and the nonexistence,the uniqueness and the energy decay estimate of solution for the fourth-order nonlinear wave equation utt＋αΔ2 u-bΔut-βΔu＋ut｜ut｜^r＋g（u）=0 in Ω×（0,∞） are studied with the boundary condition u=（u）/（υ）=0 onΩ and the initial condition u（x,0）=u0（x）,ut（x,0）=u1（x,0） in bounded domain ΩR^n ,n≥1.The energy decay rate of the global solution is estimated by the multiplier method.The blow-up result of the solution in finite time is established by the ideal of a potential well theory,and the existence of the solution is gotten by the Galekin approximation method.

SOME RESULTS ON HYPERBOLIC SYSTEMS WITH RELAXATION

In this article, first, the authors prove that there exists a unique global smooth solution for the Cauthy problem to the hyperbolic conservation laws systems with relaxation; second, in the large time station, they prove that the global smooth solutions of the hyperbolic conservation laws systems with relaxation converge to rarefaction waves solution at a determined L^P（p ≥ 2） decay rate.

Well-posedness for A Plate Equation with Nonlocal Source term

In this paper,we investigate the initial boundary value problem for a plate equation with nonlocal source term.The local,global existence and exponential decay result are established under certain conditions.Moreover,we also prove the blow-up in finite time and the lifespan of solution under certain conditions.

Spatial Decay Estimates for the Moore-Gibson-Thompson Heat Equation

In this article,the Moore-Gibson-Thompson heat equation in three-dimensional cylindrical domain are studied.Using a second order differential inequality,we obtain that the solution can decay exponentially as the distance from the entry section tends to infinity.Our result can be seen as a version of Saint-Venant principle.

A Logarithmic Decay of the Energy for the Hyperbolic Equation with Supercritical Damping

We are concerned with the following quasilinear wave equation involving variable sources and supercritical damping:■Generally speaking,when one tries to use the classical multiplier method to analyze tRhe asymptotic behavior of solutions,an inevitable step is to deal with the integralΩ|ut|^(m−2)utudx.A usual technique is to apply Young’s inequality and Sobolev embedding inequality to use the energy function and its derivative to control this integral for the subcritical or critical damping.However,for the supercritical case,the failure of the Sobolev embedding inequality makes the classical method be impossible.To do this,our strategy is to prove the rate of the integral RΩ|u|^(m)dx grows polynomially as a positive power of time variable t and apply the modified multiplier method to obtain the energy functional decays logarithmically.These results improve and extend our previous work[12].Finally,some numerical examples are also given to authenticate our results.

Blow-Up of Solution and Energy Decay for a Quasilinear Parabolic Problem

In this paper,we obtain the blow-up result of solutions and some general decay rates for a quasilinear parabolic equation with viscoelastic terms A(t)|u_(t)|^(m-2)u_(t)-△u+∫_(0)^(t)g(t-s)△u(t-s)△(x,s)ds=|u|^(p-2)ulog|u|.Due to the presence of the log source term,it is not possible to use the source term to dominate the term A(t)|u_(t)|^(m-2)u_(t).To bypass this difficulty,we build up inverse Holder-like inequality and then apply differential inequality argument to prove the solution blows up in finite time.in addition,we can also give a decay rate under a general assumption on the relaxation functions satisfying g′≤-ζ(t)H(g(t),H(t))=t^(v),t≥0,v>1.This improves the existing results.

Global existence and decay of smooth solutions for the 3-D MHD-type equations without magnetic diffusion

We study the large time behavior of a 3-D MHD(magneto-hydrodynamical)-type system without magnetic diffusion introduced by Lin and Zhang(2014). By using the elementary energy method and interpolation technique, we prove the global existence and decay estimate of smooth solution near the equilibrium state(x3, 0).

Exponential Decay of Energy for a Logarithmic Wave Equation

In this paper we consider the initial boundary value problem for a class of logarithmic wave equation. By constructing an appropriate Lyapunov function, we obtain the decay estimates of energy for the logarithmic wave equation with linear damping and some suitable initial data. The results extend the early results.

Global Existence and Decay Estimate of Solution to One Dimensional Convection-Diffusion Equation

We study the global existence of solution to one di- mensional convection-diffusion equation. Through constructing a Cauchy sequence in a Banach space, we get the local existence of solution to the equation, t3ased on the global bounds of the solu- tion, we extend the local one to a global one that decays in Hl space.

OPTIMAL DECAY RATE OF THE COMPRESSIBLE QUANTUM NAVIER-STOKES EQUATIONS

For quantum fluids governed by the compressible quantum Navier-Stokes equations in R;with viscosity and heat conduction, we prove the optimal L;- L;decay rates for the classical solutions near constant states. The proof is based on the detailed linearized decay estimates by Fourier analysis of the operators, which is drastically different from the case when quantum effects are absent.

Decay estimates of discretized Green's functions for Schrdinger type operators

For a sparse non-singular matrix A, generally A- 1 is a dense matrix. However, for a class of matrices, A-1 can be a matrix with off-diagonal decay properties, i.e., ｜Aij^-1｜ decays fast to 0 with respect to the increase of a properly defined distance between i and j. Here we consider the off-diagonal decay properties of discretized Green＇s functions for SchrSdinger type operators. We provide decay estimates for discretized Green＇s functions obtained from the finite difference discretization, and from a variant of the pseudo-spectral discretization. The asymptotic decay rate in our estimate is independent of the domain size and of the discretization parameter. We verify the decay estimate with numerical results for one-dimensional Schr6dinger type operators.