In this paper, we consider a coupled Lamé system with a viscoelastic damping 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 introducing a suitable Lyapunov functional.

The initial-boundary value problem of an anisotropic porous medium equation $$u_t=\sum^N_{i=1}\frac{\partial}{\partial x_i}(a(x,t)|u|^{\alpha_i}u_{x_i})+\sum^N_{i=1}\frac{\partial f_i(u,x,t)}{\partial x_i}$$ is studied. Compared with the usual porous medium equation, there are two different 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 boundedness of the gradient $|∇u|$ for the usual porous medium, we can only show that $∇u∈ L^∞(0,T;L^2_{ {\rm loc}}(Ω)).$ 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.

We are concerned with the following quasilinear wave equation involving variable sources and supercritical damping: $$u_{tt}-{\rm div}(|\nabla u|^{p(x)-2}\nabla u)+|u_t|^{m-2}u_t=|u|^{q(x)-2}u.$$ 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 $\int_Ω |u_t|^{m−2}u_tudx.$ 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 $\int_Ω |u|^ mdx$ 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.

In this paper, we consider the Cauchy problem of a multi-dimensional radiating 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 nuclear phenomena. One of our main motivations is to attempt to explore how nonlinear radiative inhomogeneity influences the behavior of entropy solutions. Simple but different 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 scaling. 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.

This article investigates the blow-up results for the initial boundary value problem to the quasi-linear parabolic equation with $p$-Laplacian $$u_t−∇·( |∇u|^{p−2}∇u)= f(u),$$ where $p≥2$ and the function $f(u)$ satisfies $$α\int^u_0f(s)ds≤u f(u)+βu^p+\gamma, u>0$$ for some positive constants $α,β,\gamma$ with $0<β≤ \frac{(α−p)λ_{1,p}}{p},$ which has been studied under the initial condition $J_p(u_0)<0.$ This paper generalizes the above results on the following 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.

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 coefficients 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 $W^{m,p}$-regularity to the $W^{m,p(t,x)}$-regularity for such higher-order parabolic equations with merely measurable coefficients with Reifenberg flat domain which is beyond the Lipschitz domain with small Lipschitz constant.