This paper deals with parabolic-elliptic chemotaxis systems with the sensitivity function $\chi \left(v\right)$ and the growth term $f\left(u\right)$ under homogeneous Neumann boundary conditions in a smooth bounded domain. Here it is assumed that $0<\chi \left(v\right)\le {\chi }_0/v^k$$(k\ge 1...$

A necessary and sufficient condition for the boundedness of a solution of the third problem for the Laplace equation is given. As an application a similar result is given for the third problem for the Poisson equation on domains with Lipschitz boundary.

In this paper we derive lower bounds and upper bounds on the effective properties for nonlinear heterogeneous systems. The key result to obtain these bounds is to derive a variational principle, which generalizes the variational principle by P. Ponte Castaneda from 1992. In general, when the Ponte Castaneda variational principle is used one only gets either a lower or an upper bound depending on the growth conditions. In this paper we overcome this problem by using our new variational principle...

In this paper we derive upper and lower bounds on the homogenized energy density functional corresponding to degenerated $p$-Poisson equations. Moreover, we give some non-trivial examples where the bounds are tight and thus can be used as good approximations of the homogenized properties. We even present some cases where the bounds coincide and also compare them with some numerical results.

We consider the cubic Nonlinear Schrödinger Equation (NLS) and the Korteweg-de Vries equation in one space dimension. We prove that the solutions of NLS satisfy a-priori local in time $H^s$ bounds in terms of the $H^s$ size of the initial data for $s\ge -\frac{1}{4}$ (joint work with D. Tataru, [15, 14]) , and the solutions to KdV satisfy global a priori estimate in $H^{-1}$ (joint work with T. Buckmaster [2]).

We consider a special type of a one-dimensional quasilinear wave equation wtt - phi (wt / wx) wxx = 0 in a bounded domain with Dirichlet boundary conditions and show that classical solutions blow up in finite time even for small initial data in some norm.

The semilinear differential equation (1), (2), (3), in $ℝ\times \mathrm{\Omega }$ with $\mathrm{\Omega }\in {ℝ}^N$, (a nonlinear wave equation) is studied. In particular for $\mathrm{\Omega }={ℝ}^3$, the existence is shown of a weak solution $u(t,x)$, periodic with period $T$, non-constant with respect to $t$, and radially symmetric in the spatial variables, that is of the form $u(t,x)=\nu (t,|x|)$. The proof is based on a distributional interpretation for a linear equation corresponding to the given problem, on the Paley-Wiener criterion for the Laplace Transform, and on the alternative method of...

Let $\Omega$ be a bounded domain in ${ℝ}^n$ with smooth boundary $\partial \Omega$. We consider the equation $d^2\Delta u-u+u^{\frac{n-k+2}{n-k-2}}=0\text{in}\Omega$, under zero Neumann boundary conditions, where $\Omega$ is open, smooth and bounded and $d$ is a small positive parameter. We assume that there is a $k$-dimensional closed, embedded minimal submanifold $K$ of $\partial \Omega$, which is non-degenerate, and certain weighted average of sectional curvatures of $\partial \Omega$ is positive along $K$. Then we prove the existence of a sequence $d=d_j\to 0$ and a positive solution $u_d$ such that $d^2{\left|\nabla u_d\right|}^2\rightharpoonup S{\delta }_K\mathrm{as}d\to 0$ in the sense of measures, where ${\delta }_K$...

The object of this paper is to find the solution of the differential equation of the buckling problem of anisotropic cylindrical shells with shear load in case of torsion of a long tube. The critical values of the shear load and the total torque are also found. The corresponding results for the isotropic case are deduced as a special case.