Exact Non-Self-Similar Solutions of the Equation $$u_t = \Delta \ln u$$

In this paper, we obtain new exact non-self-similar solutions of the nonlinear diffusion equation $$\begin{gathered} {\text{ }}u_t = \Delta \ln u, \hfill \\ u \triangleq u\left( {x,t} \right):\Omega \times \mathbb{R}^ + \to \mathbb{R},{\text{ }} x \in \mathbb{R}^n , \hfill \\ \end{gathered} $$ where $\Omega \subset \mathbb{R}^n $ is the domain and $\mathbb{R}^ + = \left\{ {t:0 \leqslant t < + \infty } \right\},{\text{ }}u\left( {x,t} \right) \geqslant 0$ is the temperature of the medium.     

Some approximation theorems for modified Bernstein-Durrmeyer operators

The modified Bernstein-Durrmeyer operators discussed in this paper are given byM_nf≡M_n(f,x)=(n+2)P_(n,k)∫_0~1p_n+1.k(t)f(t)dt,whereWe will show,for 0<α<1 and 1≤p≤∞     

Nonlinear wavelet methods for high-dimensional backward heat equation

The backward heat equation is a typical ill-posed problem. In this paper, we shall apply a dual least squares method connecting Shannon wavelet to the following equation ut (x, y, t) = u xx (x, y, t) + uyy (x, y, t), x ∈ R, y ∈ R, 0 ≤ t 1, u(x, y, 1) = (x, y), x ∈ R, y ∈ R. Motivated by Regińska's work, we shall give two nonlinear approximate methods to regularize the approximate solutions for high-dimensional backward heat equation, and prove that our methods are convergent.     

Bang-bang property for Bolza problems in two dimensions

Consider the following Bolza problem: $$\begin{gathered} \min \int {h(x,u) dt,} \hfill \\ \dot x = F(x) + uG(x), \hfill \\ \left| u \right| \leqslant 1, x \in \Omega \subset \mathbb{R}^2 , \hfill \\ x(0) = x_0 , x(1) = x_1 . \hfill \\ \end{gathered} $$ We show that, under suitable assumptions onF, G, h, all optimal trajectories are bang-bang. The proof relies on a geometrical approach that works for every smooth two-dimensional manifold. As a corollary, we obtain existence results for nonconvex optimization problems.     

The critical exponent for a weakly coupled system of the generalized Fujita type reaction-diffusion equations

We study nonnegative solutions of the initial value problem for a weakly coupled system

Nontrivial solution of a quasilinear elliptic equation with critical growth in ℝn

Suppose Δ_n u = div (¦ ?u ¦^n-2?u) denotes then-Laplacian. We prove the existence of a nontrivial solution for the problem $$\left\{ \begin{gathered} - \Delta _n u + \left| u \right|^{n - 2} u = \int {(x,u)u^{n - 2} in \mathbb{R}^n } \hfill \\ u \in W^{1,n} (\mathbb{R}^n ) \hfill \\ \end{gathered} \right.$$ wheref(x, t) =o(t) ast → 0 and ¦f(x, t)¦ ≤C exp(α_n¦t¦^n/(n-1)) for some constantC > 0 and for allx∈?;t∈? with α_n =nω _n ^1/(n-1) , ω_n = surface measure ofS ^n-1.     

Sharpness on the Lower Bound of the Lifespan of Solutions to Nonlinear Wave Equations

This paper is devoted to proving the sharpness on the lower bound of the lifespan of classical solutions to general nonlinear wave equations with small initial data in the case n = 2 and cubic nonlinearity (see the results of T. T. Li and Y. M. Chen in 1992). For this purpose, the authors consider the following Cauchy problem:

Stability of sets for impulsive functional differential equations via Razumikhin method

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

Symmetry results for systems involving fractional Laplacian

In this paper we investigate symmetry results for positive solutions of systems involving the fractional Laplacian (1) $\left\{ \begin{gathered} ( - \Delta )^{\alpha _1 } u_1 (x) = f_1 (u_2 (x)),x \in \mathbb{R}^\mathbb{N} , \hfill \\ ( - \Delta )^{\alpha _2 } u_2 (x) = f_2 (u_1 (x)),x \in \mathbb{R}^\mathbb{N} , \hfill \\ \lim _{|x| \to \infty } u_1 (x) = \lim _{|x| \to \infty } u_2 (x) = 0 \hfill \\ \end{gathered} \right. $ where N ≥ 2 and α ₁, α ₂ ∈ (0, 1). We prove symmetry properties by the method of moving planes.     

On boundary values of solutions of the dirichlet problem for second order elliptic equation

The paper suggests some conditions on the lower order terms, which provide that the solution of the Dirichlet problem for the general elliptic equation of the second order

Some dichotomy results for the quenching problem

It is shown that any solution to the semilinear problem{ ut = uxx + δ(1-u)-p , (x, t) ∈ (-1 , 1) × (0 , T ), u( ±1 , t) = 0, t ∈ (0 , T ), u(x, 0) = u0(x) 1, x ∈ [ 1 , 1] either touches 1 in finite time or converges smoothly to a steady state as t →∞. Some extensions of this result to higher dimensions are also discussed.     

Finite difference method of first boundary problem for quasilinear parabolic systems (IV)

More work is done to study the explicit, weak and strong implicit difference solution for the first boundary problem of quasilinear parabolic system: $$\begin{gathered} u_t = ( - 1)^{M + 1} A(x,t,u, \cdots ,u_x M - 1)u_x 2M + f(x,t,u, \cdots u_x 2M - 1), \hfill \\ (x,t) \in Q_T = \left| {0< x< l,0< t \leqslant T} \right|, \hfill \\ u_x ^k (0,t) = u_x ^k (l,t) = 0 (k = 0,1, \cdots ,M - 1),0< t \leqslant T, \hfill \\ u(x,0) = \varphi (x),0 \leqslant x \leqslant l, \hfill \\ \end{gathered} $$ whereu, ?, andf arem-dimensional vector valued functions, A is anm×m positively definite matrix, and $u_t = \frac{{\partial u}}{{\partial t}},u_x ^k = \frac{{\partial ^k u}}{{\partial x^k }}$ . For this problem, the convergence of iteration for the general difference schemes is proved.     

Blow-up,quenching, aggregation and collapse in a chemotaxis model with reproduction term

In this paper,we consider the following chemotaxis model with ratio-dependent logistic reaction term u/t=D▽(▽u-u▽ω/ω)+u(α-bu/ω),(x,t)∈QT,ω/t=βu-δω,(x,t)∈QT,u▽㏑(u/w)·=0,x ∈Ω,0tT,u(x,0)=u0(x)0,x ∈,w(x,0)=w0(x)0,x ∈,It is shown that the solution to the problem exists globally if b+β≥0 and will blow up or quench if b+β0 by means of function transformation and comparison method.Various asymptotic behavior related to different coefficients and initial data is also discussed.     

Mean value theorem and a maximum principle for Kolmogorov's equation

For an equation of the form $$\begin{gathered} \frac{{\partial u}}{{\partial t}} - \sum\nolimits_{ij = 1}^n {{\text{ }}\alpha ^{ij} } \frac{{\partial ^2 u}}{{\partial x^i \partial x^j }} + \sum\nolimits_{ij = 1}^n {\beta _j^i x^i } \frac{{\partial u}}{{\partial x^i }} = 0, \hfill \\ {\text{ }}x \in R^n ,{\text{ }}t \in R^1 , \hfill \\ \end{gathered}$$ where α=(α^ij) is a constant nonnegative matrix andΒ=(Β _i ⁱ ) is a constant matrix, subject to certain conditions, we construct a fundamental solution, similar in its structure to the fundamental solution of the heat conduction equation; we prove a mean value theorem and show that u(x₀, t₀) can be represented in the form of the mean value of u(x, t) with a nonnegative density over a level surface of the fundamental solution of the adjoint equation passing through the point (x₀, t₀); finally, we prove a parabolic maximum principle.     

Oscillation of solutions of nonlinear partial differential equations of neutral type

In this paper, we deal with the oscillatory behavior of solutions of the neutral partial differential equation of the form $$\begin{gathered} \frac{\partial }{{\partial t}}\left[ {p\left( t \right)\frac{\partial }{{\partial t}}(u\left( {x,t} \right) + \sum\limits_{i = 1}^t {p_i \left( t \right)u\left( {x,t - \tau _i } \right)} )} \right] + q\left( {x,t} \right)f_j (u(x,\sigma _j (t))) \hfill \\ = a\left( t \right)\Delta u\left( {x,t} \right) + \sum\limits_{k = 1}^n {a_k \left( t \right)} \Delta u\left( {x,\rho _k \left( t \right)} \right), \left( {x,t} \right) \in \Omega \times R_ + \equiv G \hfill \\ \end{gathered} $$ where Δ is the Laplacian in EuclideanN-spaceR ^N, R₊=(0, ∞) and Ω is a bounded domain inR ^N with a piecewise smooth boundary δΩ.     

On the globality of a solution of one system of Schrödinger equations

A system of nonlinear Schrödinger equations $\begin{gathered} \frac{{\partial u_k }}{{\partial t}} = ia_k \Delta u_k + f_k (u,u^* ), t > 0, k = 1,...,m, \hfill \\ u_k (0,x) = u_{k0} (x), k = 1,...,m, x \in R^n . \hfill \\ \end{gathered} $ is investigated. Conditions that assure the globality of a solution are found.     

Heat equation with a singular potential on the boundary and the Kato inequality

This paper is concerned with the heat equation in the half-space ? ₊ ^N with the singular potential function on the boundary, (P) $\left\{ \begin{gathered} \frac{\partial } {{\partial t}}u - \Delta u = 0\operatorname{in} \mathbb{R}_ + ^N \times (0,T), \hfill \\ \frac{\partial } {{\partial x_N }}u + \frac{\omega } {{|x|}}u = 0on\partial \mathbb{R}_ + ^N \times (0,T), \hfill \\ u(x,0) = u_0 (x) \geqslant ()0in\mathbb{R}_ + ^N , \hfill \\ \end{gathered} \right. $ where N ?? 3, ?? > 0, 0 < T ?? ??, and u ₀ ?? C ₀(? ₊ ^N ). We prove the existence of a threshold number ?? _N for the existence and the nonexistence of positive solutions of (P), which is characterized as the best constant of the Kato inequality in ? ₊ ^N .     

Stability of the Faedo-Galerkin approximation of nonlinear wave-equations

Present investigation analyses the Ljapunov stability of the systems of ordinary differential equations arising in then-th step of the Faedo-Galerkin approximation for the nonlinear wave-equation $$\begin{gathered} u_{tt} - u_{xx} + M(u) = 0 \hfill \\ u(0,t) = u(1,t) = 0 \hfill \\ u(x,0) = \Phi (x); u_t (x,0) = \Psi (x). \hfill \\ \end{gathered}$$ For the nonlinearities of the classM (u)=u ^{2 p+1} ,p ∈N, ann-independent stability result is given. Thus also the stability of the original equation is shown.     

Existence and iteration of positive solution for a three-point boundary value problem with ap-Laplacian operator

In the paper, we obtain the existence of positive solutions and establish a corresponding iterative scheme for BVPs $$\left\{ \begin{gathered} (\phi _p (u\prime ))\prime + q(t)f(t, u) = 0,0< t< 1, \hfill \\ u(0) - B(u\prime (\eta )) = 0, u\prime (1) = 0 \hfill \\ \end{gathered} \right.$$ and $$\left\{ \begin{gathered} (\phi _p (u\prime ))\prime + q(t)f(t, u) = 0,0< t< 1, \hfill \\ u\prime (0) = 0, u(1) + B(u\prime (\eta )) = 0 \hfill \\ \end{gathered} \right.$$ The main tool is the monotone iterative technique. Here, the coefficientq(t) may be singular att = 0,1.     

Numerical blow-up for a nonlinear heat equation

This paper concerns the study of the numerical approximation for the following initialboundary value problem