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Publications (39)
This paper is concerned with energy decay rates for wave systems with indirect time-space dependent damping. With the help of the method of weighted energy integral inequalities, the explicit decay rates depending on the damping coefficient are obtained. Our results are a generalization of Alabau-Boussouira, Wang, Yu(ESAIM: COCV, 23 (2017) 721–749....
A degenerate wave equation with time-varying delay in the boundary control input is considered. The well-posedness of the system is established by applying the semigroup theory. The boundary stabilization of the degenerate wave equation is concerned and the uniform exponential decay of solutions is obtained by combining the energy estimates with su...
In this paper, a nonlinearly damped system of wave equations is considered. Uniform energy decay was discussed in the previous work (Discrete Contin. Dyn. Syst. Ser. S, 2 (2009) 583–608) for if the space dimension is 3. New energy decay is proposed for by choosing appropriate multiplier related to a non‐increasing differential function. As an examp...
In this paper, the Cauchy problem for the nonlinear Schrödinger system
\begin{document}$ \begin{equation*} \begin{cases} i\partial_tu_1(x, t) = \Delta u_1(x, t)-|u_1(x, t)|^{p-1}u_1(x, t)-|u_2(x, t)|^{p-1}u_1(x, t), \\ i\partial_tu_2(x, t) = \Delta u_2(x, t)-|u_2(x, t)|^{p-1}u_2(x, t)-|u_1(x, t)|^{p-1}u_2(x, t), \end{cases} \end{equation*} $\end{do...
In this paper, the Cauchy problem for a class of coupled system of the four-dimensional cubic focusing nonlinear Schrödinger equations was investigated. By exploiting the double Duhamel method and the long-time Strichartz estimate, the global well-posedness and scattering were proven for the system below the ground state. In our proof, we first est...
In this paper, a class of damped viscoelastic wave equations
is considered in a bounded domain \(\Omega \subset {\mathbb {R}}^3\). Uniform energy decay was discussed which depends on the relaxation function \(-k'(s)\) in the previous work (Guo et al., Z Angew Math Phys 69:65, 2018) for \(1\le m\le 5\). Depending on a key integral inequality obtaine...
In this paper, a class of viscoelastic evolution equation on manifolds with conical singularities is concerned. Local existence and uniqueness of the weak solution are established by making use of the Faedo-Galerkin method and the contraction mapping principle. Moreover, by constructing an auxiliary functional and exploiting Levine’s concavity meth...
The purpose of this paper is to study the following equation driven by a nonlocal integro‐differential operator LK$$ {\mathcal{L}}_K $$: utt+[u]s2(θ−1)LKu+a|ut|m−1ut=b|u|p−1u,$$ {u}_{tt}+{\left[u\right]}_s^{2\left(\theta -1\right)}{\mathcal{L}}_Ku+a{\left|{u}_t\right|}^{m-1}{u}_t=b{\left|u\right|}^{p-1}u, $$ with homogeneous Dirichlet boundary cond...
In this paper, a class of variable-coefficient wave equations equipped with time-dependent damping and the nonlinear source is considered. We show that the total energy of the system decays to zero with an explicit and precise decay rate estimate under different assumptions on the feedback with the help of the method of weighted energy integral.
In this note, we investigate again the blow-up phenomenon of weak solutions to the following initial-boundary value problem of the fourth-order equation with variable-exponent nonlinearity utt+Δ2u-M(‖∇u‖22)Δu-Δut+|ut|m(x)-2ut=|u|p(x)-2u.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb...
In this paper, a singular non-Newton polytropic filtration equation under the initial-boundary value condition is revisited. The finite time blow-up results were discussed when the initial energy E(u0) was subcritical (E(u0)<d), critical (E(u0)=d), and supercritical (E(u0)>d), with d being the potential depth by using the potential well method and...
The goal of the present paper is to study the viscoelastic wave equation with the time-varying delay under initial-boundary value conditions. By using the multiplier method together with some properties of the convex functions, the explicit and general stability results of the total energy are proved under the general assumption on the relaxation f...
In this manuscript we consider a coupled, by second order terms, system of two wave equations with a past history acting on the first equation as a stabilizer. We show that the solution of this system decays exponentially by constructing an appropriate Lyapunov function.
In this paper, a system of coupled quasi-linear and linear wave equations with a finite memory term is concerned. By constructing an appropriate Lyapunov function, we prove that the total energy associated with the system is stable under suitable conditions on memory kernel.
The aim of this paper is to apply the modified potential well method and some new differential inequalities to study the asymptotic behavior of solutions to the initial homogeneous Neumann problem of a nonlinear diffusion equation driven by the p(x)-Laplace operator. Complete classification of global existence and blow-up in finite time of solution...
This paper deals with the following Petrovsky equation with damping and nonlinear sources: utt+Δ2u−M(‖∇u‖22)Δu−Δut+|ut|m(x)−2ut=|u|p(x)−2u under initial-boundary value conditions, where M(s) = a + bsγ is a positive C1 function with the parameters a > 0, b > 0, γ ⩾ 1, and m(x) and p(x) are given measurable functions. The upper bound of the blow-up t...
The purpose of this paper is to study the following equation driven by a nonlocal integro-differential operator $\mathcal{L}_K$: \[u_{tt}+[u]_s^{2(\theta-1)}\mathcal{L}_Ku+a|u_t|^{m-1}u_t=b|u|^{p-1}u\] with homogeneous Dirichlet boundary condition and initial data, where $[u]^2_s$ is the Gagliardo seminorm, $a\geq 0,~b>0,~0
The goal of the present paper is to study the viscoelastic wave equation with variable exponents $$\begin{aligned} u_{tt}-\Delta _{p(x)}u-\Delta u+\int _0^tg(t-s)\Delta u(s)\mathrm{{d}}s-\Delta u_t=|u|^{q(x)-2}u \end{aligned}$$under initial-boundary value conditions, where the exponents of nonlinearity p(x) and q(x) are given functions. To be more...
This paper deals with the following Petrovsky equation with damping and nonlinear source \[u_{tt}+\Delta^2 u-M(\|\nabla u\|_2^2)\Delta u-\Delta u_t+|u_t|^{m(x)-2}u_t=|u|^{p(x)-2}u\] under initial-boundary value conditions, where $M(s)=a+ bs^\gamma$ is a positive $C^1$ function with parameters $a>0,~b>0,~\gamma\geq 1$, and $m(x),~p(x)$ are given mea...
The goal of the present paper is to study the viscoelastic wave equation with the time delay \[ |u_t|^\rho u_{tt}-\Delta u-\Delta u_{tt}+\int_0^tg(t-s)\Delta u(s)ds+\mu_1u_t(x,t)+\mu_2 u_t(x,t-\tau)=b|u|^{p-2}u\] under initial boundary value conditions, where $\rho,~b,~\mu_1$ are positive constants, $\mu_2$ is a real number, $\tau>0$ represents the...
This paper deals with a pseudo‐parabolic equation with nonstandard growth conditions. Some results on global existence, blow‐up, and asymptotic behavior of weak solutions were obtained. The purpose of this paper is to give results to some unsolved questions. First, the precise decay rates for global solutions are derived. Second, an explicit upper...
This short note is concerned with a quasilinear diffusion equation under initial and Neumann boundary value condition. To be more precise, the authors establish a gradient maximum principle of classical solutions via the maximum principle and Hopf’s lemma. The result generalizes a recent work obtained by Kim (Proc Amer Math Soc 145:1203–1208, 2017)...
This paper deals with the following viscoelastic wave equation with a strong damping and logarithmic nonlinearity:
\begin{document}$ u_{tt}-\Delta u+\int_0^tg(t-s)\Delta u(s)ds-\Delta u_t = |u|^{p-2}u\ln|u|. $\end{document}
A finite time blow-up result is proved for high initial energy. Meanwhile, the lifespan of the weak solution is discussed. The...
The aim of this paper is to study bounds for blow-up time to the following viscoelastic hyperbolic equation of Kirchhoff type with initial-boundary value condition: |ut|ρutt−M(∥∇u∥22)Δu+∫0tg(t−τ)Δu(τ)dτ+|ut|m(x)−2ut=|u|p(x)−2u. Compared with constant exponents, it is difficult to discuss the above problem due to the existence of a gap between the m...
The goal of the present paper is to study the asymptotic behavior of solutions for the viscoelastic wave equation with variable exponents \[ u_{tt}-\Delta u+\int_0^tg(t-s)\Delta u(s)ds+a|u_t|^{m(x)-2}u_t=b|u|^{p(x)-2}u\] under initial-boundary condition, where the exponents $p(x)$ and $m(x)$ are given functions, and $a,~b>0$ are constants. More pre...
In this paper, a class of non-Newton filtration equations with singular potential and logarithmic nonlinearity under initial-boundary condition is investigated. Based on potential well method and Hardy-Sobolev inequality, the global existence of solutions is derived when the initial energy $J(u_0)$ is subcritical($J(u_0)<d$), critical($J(u_0)=d$) w...
The aim of this paper is to apply the modified potential well method and some new differential inequalities to study the asymptotic behavior of solutions to the initial homogeneous $\hbox{Neumann}$ problem of a nonlinear diffusion equation driven by the $p(x)$-\hbox{Laplace} operator. Complete classification of global existence and blow-up in finit...
In this paper, a doubly nonlinear parabolic problem involving nonstandard growth conditions under homogeneous Dirichlet boundary conditions is studied. To be a little precise, first, some energy estimates and inequalities are exploited to obtain energy decay estimates of the solution. Then, it is proved that the rest field u(x, t) = 0 is asymptotic...
In this paper, we study the fractional p -Laplacian evolution equation with arbitrary initial energy,
$$\begin{array}{} \displaystyle u_t(x,t) + (-{\it\Delta})_p^s u(x,t) = f(u(x,t)), \quad x\in {\it\Omega}, \,t \gt 0, \end{array} $$
where $\begin{array}{} (-{\it\Delta})_p^s \end{array} $ is the fractional p -Laplacian with $\begin{array}{} p \gt \...
This paper deals with a pseudo-parabolic equation involving variable exponents under initial and Dirichlet boundary value conditions. We obtain the global existence and blow-up results of weak solutions with arbitrarily high initial energy. These results extend and improve some recent results in which the blow-up results were showed when the initia...
This paper deals with the following pseudo‐parabolic equation ut−Δut−div(|∇u|m(x)−2∇u)=|u|p(x)−2u under initial and Dirichlet boundary value conditions. The authors establish some qualitative relationships by constructing a new control function and applying the Sobolev embedding inequality, and then decay estimates are obtained due to a key integra...
In this paper, we consider the following quasilinear hyperbolic equation involving variable sources: utt−div(|∇u|p(x)−2∇u)+|ut|m(x)−2ut=|u|q(x)−2u.Some energy estimates and Komornik inequality are used to prove some uniform estimates of decay rates of the solution. And then, we prove that u(x,t)=0 is asymptotic stable in terms of natural energy ass...
This paper deals with a pseudo-parabolic equation involving variable exponents under Dirichlet boundary value condition. The author proves that the solution is not global in time when the initial energy is positive. This result extends and improves a recent result obtained by Di et al. (2017) [1].
We study the closure of approximating sequences of some diffusion equations under certain weak convergence. A specific description of the closure under weak H 1 H^1 -convergence is given, which reduces to the original equation when the equation is parabolic. However, the closure under strong L 2 L^2 -convergence may be much larger, even for parabol...
We study the closure of approximating sequences of some diffusion equations under certain weak convergence. A specific description of the closure under weak $H^1$-convergence is given, which reduces to the original equation when the equation is parabolic. However, the closure under strong $L^2$-convergence may be much larger, even for parabolic equ...
This paper is concerned with the blow-up of solutions to the following nonlocal p-Laplace equation:
$$u_t-\mathrm{div}(|\nabla{u}|^{p-2}\nabla{u})=|u|^{q-1}u-\frac{1}{|\Omega|} \int\limits_\Omega{|u|^{q-1}u}dx,\quad x\in\Omega,\quad 0 < t < T,
$$under homogeneous Neumann boundary conditions in a bounded smooth domain \({\Omega\subset\mathrm{R}^N}\)...
