Asymptotics of solutions to the periodic problem for the nonlinear damped wave equation

We study large time asymptotic behavior of solutions to the periodic problem for the nonlinear damped wave equation

Stability of sets for impulsive functional differential equations via Razumikhin method

In this paper, we consider the functional differential equation with impulsive perturbations

A note on a fourth order degenerate parabolic equation in higher space dimensions

We are concerned with existence, positivity property and long-time behavior of solutions to the following initial boundary value problem of a fourth order degenerate parabolic equation in higher space dimensions      

A relaxation result for micromagnetics in SBV

An integral representation for the functional

Asymptotic profile of solutions to nonlinear dissipative evolution system with ellipticity

We consider the Cauchy problem for the nonlinear dissipative evolution system with ellipticity on one dimensional space

Asymptotic behavior of positive solutions of a Dirichlet problem involving combined nonlinearities

We study the behavior of positive solutions of the following Dirichlet problem

Symmetry-Breaking Phenomena in an Optimization Problem for some Nonlinear Elliptic Equation

Let $\Omega$ be a bounded domain in ${\bf R^n}$ with Lipschitz boundary, $\lambda >0,$ and $1\le p \le (n+2)/(n-2)$ if $n\ge 3$ and $1\le p< +\infty$ if $n=1,2$. Let $D$ be a measurable subset of $\Omega$ which belongs to the class $ {\cal C}_{\beta}=\{D\subset \Omega \quad | \quad |D|=\beta\} $ for the prescribed $\beta\in (0, |\Omega|).$ For any $D\in{\cal C}_{\beta}$, it is well known that there exists a unique global minimizer $u\in H^1_0(\Omega)$, which we denote by $u_D$, of the functional \[\quad J_{\Omega,D}(v)=\frac12\int_{\Omega}|\nabla v|^2\, dx+\frac{\lambda}{p+1}\int_{\Omega}|v|^{p+1}\, dx -\int_{\Omega}\chi_Dv\,dx \] on $H^1_0(\Omega)$. We consider the optimization problem $ E_{\beta,\Omega}=\inf_{D\in {\cal C}_{\beta}} J_D(u_D) $ and say that a subset $D^*\in {\cal C}_{\beta}$ which attains $E_{\beta,\Omega}$ is an optimal configuration to this problem. In this paper we show the existence, uniqueness and non-uniqueness, and symmetry-preserving and symmetry-breaking phenomena of the optimal configuration $D^*$ to this optimization problem in various settings.     

Oscillation and Asymptotic Behavior for <Emphasis Type="Italic">n</Emphasis>th-order Nonlinear Neutral Delay Dynamic Equations on Time Scales

In this paper, we derive some sufficient conditions for the oscillation and asymptotic behavior of the nth-order nonlinear neutral delay dynamic equations

Infinitely many sign-changing solutions of an elliptic problem involving critical Sobolev and Hardy–Sobolev exponent

We study the existence and multiplicity of sign-changing solutions of the following equation

$$\begin{array}{@{}rcl@{}} \left\{\begin{array}{lllllllll} -{\Delta} u = \mu |u|^{2^{\star}-2}u+\frac{|u|^{2^{*}(t)-2}u}{|x|^{t}}+a(x)u \quad\text{in}\, {\Omega}, \\ u=0 \quad\text{on}\quad\partial{\Omega}, \end{array}\right. \end{array} $$

where Ω is a bounded domain in \(\mathbb {R}^{N}\), 0∈?Ω, all the principal curvatures of ?Ω at 0 are negative and μ≥0, a>0, N≥7, 0<t<2, \(2^{\star }=\frac {2N}{N-2}\) and \(2^{\star }(t)=\frac {2(N-t)}{N-2}\).

     

THE FIRST BOUNDARY VALUE PROBLEM FOR SOLUTIONS OF DEGENERATE QUASUJNEAR PARABOLIC EQUATIONS

In this paper, the author proves the existence and uniqueness of nonnegative solution for the first boundary value problem of uniform degenerated parabolic equation $$\[\left\{ {\begin{array}{*{20}{c}} {\frac{{\partial u}}{{\partial t}} = \sum {\frac{\partial }{{\partial {x_i}}}\left( {v(u){A_{ij}}(x,t,u)\frac{{\partial u}}{{\partial {x_j}}}} \right) + \sum {{B_i}(x,t,u)} \frac{{\partial u}}{{\partial {x_i}}}} + C(x,t,u)u\begin{array}{*{20}{c}} {}&{(x,t) \in [0,T]} \end{array},}\{u{|_{t = 0}} = {u_0}(x),x \in \Omega ,}\{u{|_{x \in \partial \Omega }} = \psi (s,t),0 \le t \le T} \end{array}} \right.\]$$ $$\[\left( {\frac{1}{\Lambda }{{\left| \alpha \right|}^2} \le \sum {{A_{ij}}{\alpha _i}{\alpha _j}} \le \Lambda {{\left| \alpha \right|}^2},\forall a \in {R^n},0 < \Lambda < \infty ,v(u) > 0\begin{array}{*{20}{c}} {and}&{v(u) \to 0\begin{array}{*{20}{c}} {as}&{u \to 0} \end{array}} \end{array}} \right)\]$$ under some very weak restrictions, i.e. $\[{A_{ij}}(x,t,r),{B_i}(x,t,r),C(x,t,r),\sum {\frac{{\partial {A_{ij}}}}{{\partial {x_j}}}} ,\sum {\frac{{\partial {B_i}}}{{\partial {x_i}}} \in \overline \Omega } \times [0,T] \times R,\left| {{B_i}} \right| \le \Lambda ,\left| C \right| \le \Lambda ,\],\[\left| {\sum {\frac{{\partial {B_i}}}{{\partial {x_i}}}} } \right| \le \Lambda ,\partial \Omega \in {C^2},v(r) \in C[0,\infty ).v(0) = 0,1 \le \frac{{rv(r)}}{{\int_0^r {v(s)ds} }} \le m,{u_0}(x) \in {C^2}(\overline \Omega ),\psi (s,t) \in {C^\beta }(\partial \Omega \times [0,T]),0 < \beta < 1\],\[{u_0}(s) = \psi (s,0).\]$     

含有Sobolev-Hardy临界指标的奇异椭圆方程Neumann问题无穷多解的存在性

该文研究了如下的奇异椭圆方程Neumann问题$\left\{\begin{array}{ll}\disp -\Delta u-\frac{\mu u}{|x|^2}=\frac{|u|^{2^{*}(s)-2}u}{|x|^s}+\lambda|u|^{q-2}u,\ \ &;x\in\Omega,\\D_\gamma{u}+\alpha(x)u=0,&;x\in\partial\Omega\backslash\{0\},\end{array}\right.$其中$\Omega $ 是 $ R^N$ 中具有 $ C^1$边界的有界区域, $ 0\in\partial\Omega$, $N\ge5$. $2^{*}(s)=\frac{2(N-s)}{N-2}$ (该文研究了如下的奇异椭圆方程Neumann问题$\left\{\begin{array}{ll}\disp -\Delta u-\frac{\mu u}{|x|^2}=\frac{|u|^{2^{*}(s)-2}u}{|x|^s}+\lambda|u|^{q-2}u,\ \ &;x\in\Omega,\\D_\gamma{u}+\alpha(x)u=0,&;x\in\partial\Omega\backslash\{0\},\end{array}\right.$其中$\Omega $ 是 $ R^N$ 中具有 $ C^1$边界的有界区域, $ 0\in\partial\Omega$, $N\ge5$. $2^{*}(s)=\frac{2(N-s)}{N-2}$ (该文研究了如下的奇异椭圆方程Neumann问题其中Ω是RN中具有C1边界的有界区域,0∈■Ω,N≥5.2*(s)=2(N-s)/N-2(0≤s≤2)是临界Sobolev-Hardy指标, 10.利用变分方法和对偶喷泉定理,证明了这个方程无穷多解的存在性.     

单圈图的最大特征值的上界的改进

Let G（V, E） be a unicyclic graph, Cm be a cycle of length m and Cm G, and ui ∈ V（Cm）. The G - E（Cm） are m trees, denoted by Ti, i = 1, 2,..., m. For i = 1, 2,..., m, let eui be the excentricity of ui in Ti and ec = max{eui ： i = 1, 2 , m}. Let κ = ec＋1. Forj = 1,2,...,k- 1, let δij = max{dv ： dist（v, ui） = j,v ∈ Ti}, δj = max{δij ： i = 1, 2,..., m}, δ0 = max{dui ： ui ∈ V（Cm）}. Then λ1（G）≤max{max 2≤j≤k-2 （√δj-1-1＋√δj-1）,2＋√δ0-2,√δ0-2＋√δ1-1}. If G ≌ Cn, then the equality holds, where λ1 （G） is the largest eigenvalue of the adjacency matrix of G.     

Continuous dependence on the boundary conditions for the Wentzell Laplacian

The solution u of the well-posed problem

Large time behavior of small solutions to subcritical derivative nonlinear Schrödinger equations

We consider the derivative nonlinear Schrödinger equations

where the coefficient satisfies the time growth condition

is a sufficiently small constant and the nonlinear interaction term consists of cubic nonlinearities of derivative type

where and . We suppose that the initial data satifsfy the exponential decay conditions. Then we prove the sharp decay estimate , for all , where . Furthermore we show that for there exist the usual scattering states, when and the modified scattering states, when

     

On Existence and Multiplicity of Solutions for Elliptic Equations Involving the p-Laplacian

In the present paper, the following Dirichlet problem and Neumann problem involving the p-Laplacian

Optimal decay rates of solutions to dissipative nonlinear evolution equations with ellipticity

We derive the optimal decay rates of solution to the Cauchy problem for a set of nonlinear evolution equations with ellipticity and dissipative effects

On Fatou Type Convergence of Convolution Type Double Singular Integral Operators

In this paper, some approximation formulae for a class of convolution type double singular integral operators depending on three parameters of the type(T_λf)(x, y) = ∫_a~b ∫_a~b f(t, s)K_λ(t-x,s-y)dsdt, x,y ∈(a,b), λ∈Λ  [0,∞),(0.1)are given. Here f belongs to the function space L_1( a,b ~2), where a,b is an arbitrary interval in R. In this paper three theorems are proved, one for existence of the operator(T_λf)(x, y) and the others for its Fatou-type pointwise convergence to f(x_0, y_0), as(x,y,λ) tends to(x_0, y_0, λ_0). In contrast to previous works, the kernel functions K_λ(u,v)don't have to be 2π-periodic, positive, even and radial. Our results improve and extend some of the previous results of [1, 6, 8, 10, 11, 13] in three dimensional frame and especially the very recent paper [15].     

General decay and blow-up of solutions for a viscoelastic equation with nonlinear boundary damping-source interactions

In this paper, a viscoelastic equation with nonlinear boundary damping and source terms of the form $$\begin{array}{llll}u_{tt}(t)-\Delta u(t)+\displaystyle\int\limits_{0}^{t}g(t-s)\Delta u(s){\rm d}s=a\left\vert u\right\vert^{p-1}u,\quad{\rm in}\,\Omega\times(0,\infty), \\ \qquad\qquad\qquad\qquad\qquad u=0,\,{\rm on}\,\Gamma_{0} \times(0,\infty),\\ \dfrac{\partial u}{\partial\nu}-\displaystyle\int\limits_{0}^{t}g(t-s)\frac{\partial}{\partial\nu}u(s){\rm d}s+h(u_{t})=b\left\vert u\right\vert ^{k-1}u,\quad{\rm on} \ \Gamma_{1} \times(0,\infty) \\ \qquad\qquad\qquad\qquad u(0)=u^{0},u_{t}(0)=u^{1},\quad x\in\Omega, \end{array}$$ is considered in a bounded domain ??. Under appropriate assumptions imposed on the source and the damping, we establish both existence of solutions and uniform decay rate of the solution energy in terms of the behavior of the nonlinear feedback and the relaxation function g, without setting any restrictive growth assumptions on the damping at the origin and weakening the usual assumptions on the relaxation function g. Moreover, for certain initial data in the unstable set, the finite time blow-up phenomenon is exhibited.     

Strong Regularizing Effect of a Gradient Term in the Heat Equation with a Weight

We deal with the following parabolic problem, $$(P)\left\{\begin{array}{lll} u_t - \Delta{u} + |\nabla{u}|^q \quad=\quad \lambda{g}(x)u + f(x, t),\quad u > 0 \; {\rm in} \; \Omega \; \times \; (0, T),\\ \qquad\quad\quad\; u(x, t) \quad=\quad 0 \quad{\rm on}\; {\partial}{\Omega}\; \times ; (0, T),\\ \qquad\quad\quad\; u(x, 0) \quad=\quad u_{0}(x), \quad x \in {\Omega},\end{array}\right.$$ where is a bounded regular domain or ${\Omega = \mathbb{R}^N}$ , ${1 < q \leq 2, \lambda > 0\; {\rm and}\; f \geq 0, u_{0} \geq 0}$ are in a suitable class of functions. We give assumptions on g with respect to q for which for all λ >  0 and all ${f \in L^1(\Omega_T ), f \geq 0}$ , problem (P) has a positive solution. Under some additional conditions on the data, the Cauchy problem and the asymptotic behavior of the solution are also considered.     

半拟线性p-Laplacian问题结点解的存在性

In this paper, we study the existence of nodal solutions for the following problem:-(φ_p(x′))′= α(t)φ_p(x~+) + β(t)φ_p(x~-) + ra(t)f(x), 0 t 1,x(0) = x(1) = 0,where φ_p(s) = |s|~(p-2)s, a ∈ C([0, 1],(0, ∞)), x~+= max{x, 0}, x~-=- min{x, 0}, α(t), β(t) ∈C[0, 1]; f ∈ C(R, R), sf(s) 0 for s ≠ 0, and f_0, f_∞∈(0, ∞), where f_0 = lim_|s|→0f(s)/φ_p(s), f_∞ = lim|s|→+∞f(s)/φ_p(s).We use bifurcation techniques and the approximation of connected components to prove our main results.