Many authors have established results for existence and nonexistence of nonlinear evolution P.D.Es in the form $\f{\partial^k u}{\partial t^k}-\delta_1\triangle u+\delta_2\Delta^2 u=f(x,\nabla u,u)$ where $\delta_1,\delta_2\in \{0,1\}$. In this paper we will prove the nonexistence results for the general form that contains this forms for $f(x,\nabla u,u)=|x|^{-\gamma}|u|^q$ or $|u|^q$ or $|\nabla u^q|^p$.

In this paper, we show that the monotone iterative technique yields two monotone sequences that converge uniformly to extremal solutions of fourth order periodic boundary value problems and periodic solutions of functional differential equations.

A one--dimensional model for the Barkhausen effect is considered. This model describes a motion in a random medium. The motion exhibits a stick--slip type behaviour in the limit of small correlation length of the random medium. However, we prove that the velocity of the limiting motion is positive almost everywhere. For this the corresponding Fokker--Planck equation is examined. This equation is degenerated and has a critical singularity as well as no gradient structure. Therefore, the proof relies mainly on choosing the right test functions, which gives natural boundary conditions in the limit.

Motivated by questions on the preconditioning of spectral methods, and independently of the extensive literature on the approximation of zeroes of orthogonal polynomials, either by the Sturm method, or by the descent method, we develop a stationary phase-like technique for calculating asymptotics of Legendre polynomials. The difference with the classical stationary phase method is that the phase is a nonlinear function of the large parameter and the integration variable, instead of being a product of the large parameter by a function of the integration variable. We then use an implicit functions theorem for approximating the zeroes of the derivatives of Legendre polynomials. This result is used for proving order and consistency of the residual smoothing scheme [1], [19].

We use Turán type inequalities to give new non-asymptotic bounds on the extreme zeros of orthogonal polynomials in terms of the coefficients of their three term recurrence. Most of our results deal with symmetric polynomials satisfying the three term recurrence $p_{k+1}=x p_k-c_k p_{k-1},$ with a nondecreasing sequence $\{c_k\}$. As a special case they include a non-asymptotic version of Máté, Nevai and Totik result on the largest zeros of orthogonal polynomials with $c_k=c \; k^{2 \delta} (1+ o(k^{-2/3})).$ Our proof is based on new Turán inequalities which are obtained by analogy with higher order Laguerre inequalities.

In the early 1980s, Sweby [19] investigated a class of high resolution schemes using flux limiters for hyperbolic conservation laws. For the convex homogeneous conservation laws, Yang [23] has shown the convergence of the numerical solutions of semi-discrete schemes based on minmod limiter when the general building block of the schemes is an arbitrary $E$-scheme, and based on Chakravarthy-Osher limiter when the building block of the schemes is the Godunov, the Engquist-Osher, or the Lax-Friedrichs to the physically correct solution. Recently, Yang and Jiang [25] have proved the convergence of these schemes for convex conservation laws with a source term. However, the convergence problems of other flux limiter, such as van Leer and superbee have been open. In this paper, we apply the convergence criteria, established in [23] [25] by using Yang's wavewise entropy inequality (WEI) concept, to prove the convergence of the semi-discrete schemes with van Leer's limiter for the aforementioned three building blocks. The result is valid for scalar convex conservation laws in one space dimension with or without a source term. Thus, we have settled one of the aforementioned problems.