查看更多>>摘要：In this paper,we consider a coupled Lamé system with a viscoelastic damp-ing in the first equation and two strong discrete time delays.We prove its existence by using the Faedo-Galerkin method and establish an exponential decay result by intro-ducing a suitable Lyapunov functional.
查看更多>>摘要：The initial-boundary value problem of an anisotropic porous medium equa-tion ut=N∑i=1∂/∂(a(x,t)|u|αiuxi)+N∑i=1∂fi(u,x,t)/∂xi is studied.Compared with the usual porous medium equation,there are two differ-ent characteristics in this equation.One lies in its anisotropic property,another one is that there is a nonnegative variable diffusion coefficient a(x,t)additionally.Since a(x,t)may be degenerate on the parabolic boundary ∂Ω×(0,T),instead of the bound-edness of the gradient|▽u|for the usual porous medium,we can only show that▽u ∈ L∞(0,T;L2loc(Ω)).Based on this property,the partial boundary value conditions matching up with the anisotropic porous medium equation are discovered and two stability theorems of weak solutions can be proved naturally.
查看更多>>摘要：We are concerned with the following quasilinear wave equation involving variable sources and supercritical damping:utt-div(|▽u|p(x)-2▽u)+|ut|m-2ut=|u|q(x)-2u.Generally speaking,when one tries to use the classical multiplier method to analyze the asymptotic behavior of solutions,an inevitable step is to deal with the integral fΩ|ut|m-2utudx.A usual technique is to apply Young's inequality and Sobolev em-bedding 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 fΩ|u|mdx 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.
查看更多>>摘要：In this paper,we consider the Cauchy problem of a multi-dimensional radi-ating gas model with nonlinear radiative inhomogeneity.Such a model gives a good approximation to the radiative Euler equations,which are a fundamental system in radiative hydrodynamics with many practical applications in astrophysical and nu-clear phenomena.One of our main motivations is to attempt to explore how nonlinear radiative inhomogeneity influences the behavior of entropy solutions.Simple but dif-ferent phenomena are observed on relaxation limits.On one hand,the same relaxation limit such as the hyperbolic-hyperbolic type limit is obtained,even for different scal-ing.On the other hand,different relaxation limits including hyperbolic-hyperbolic type and hyperbolic-parabolic type limits are obtained,even for the same scaling if different conditions are imposed on nonlinear radiative inhomogeneity.
查看更多>>摘要：This article investigates the blow-up results for the initial boundary value problem to the quasi-linear parabolic equation with p-Laplacian ut-▽·(|▽u|p-2▽u)=f(u),where p ≥ 2 and the function f(u)satisfiesα∫u0f(s)ds≤uf(u)+βup+γ,u＞0 for some positive constants α,β,γ with 0＜β ≤(α-p)λ1,p/p,which has been studied under the initial condition Jp(uo)＜0.This paper generalizes the above results on the follow-ing aspects:a new blow-up condition is given,which holds for all p＞2;a new blow-up condition is given,which holds for p=2;some new lifespans and upper blow-up rates are given under certain conditions.
查看更多>>摘要：We prove a global estimate in the Sobolev spaces with variable exponents to the solution of a class of higher-order divergence parabolic equations with measurable coefficients over the non-smooth domains.Here,it is mainly assumed that the coef-ficients are allowed to be merely measurable in one of the spatial variables and have a small BMO quasi-norm in the other variables at a sufficiently small scale,while the boundary of the underlying domain belongs to the so-called Reifenberg flatness.This is a natural outgrowth of Dong-Kim-Zhang's papers[1,2]from the Wm,p-regularity to the Wm,p(t,x)-regularity for such higher-order parabolic equations with merely mea-surable coefficients with Reifenberg flat domain which is beyond the Lipschitz domain with small Lipschitz constant.
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