The boundary layer equations for the non-Newtonian power law fluid are examined under the classical conditions of uniform flow past a semi infinite flat plate. We investigate the behavior of the similarity solution and employing the Crocco-like transformation we establish the power series representation of the solution near the plate.

We consider the following singularly perturbed elliptic problem$$\begin{array}{cc}\hfill {ϵ}^2\Delta u-u+f\left(u\right)& =0,u\>0\text{in}\Omega ,\hfill \\ \hfill ϵ\frac{\partial u}{\partial \nu }+\lambda u& =0\text{on}\partial \Omega ,\hfill \end{array}$$where $f$ satisfies some growth conditions, $0\le \lambda \le +\infty$, and $\Omega \subset {ℝ}^N$ ($N\>1$) is a smooth and bounded domain. The cases $\lambda =0$ (Neumann problem) and $\lambda =+\infty$ (Dirichlet problem) have been studied by many authors in recent years. We show that, there exists a generic constant ${\lambda }_{*}\>1$ such that, as $ϵ\to 0$, the least energy solution has a spike near the boundary if $\lambda \le {\lambda }_{*}$, and has an interior spike near the innermost part of the domain if $\lambda \>{\lambda }_{*}$. Central to our study is the corresponding problem...

In this paper we present theoretical, computational, and practical aspects concerning 3-dimensional shape optimization governed by linear magnetostatics. The state solution is approximated by the finite element method using Nédélec elements on tetrahedra. Concerning optimization, the shape controls the interface between the air and the ferromagnetic parts while the whole domain is fixed. We prove the existence of an optimal shape. Then we state a finite element approximation to the optimization...

The paper concerns the (local and global) existence, nonexistence, uniqueness and some properties of nonnegative solutions of a nonlinear density dependent diffusion equation with homogeneous Dirichlet boundary conditions.

In this paper we consider the existence and asymptotic behavior of solutions of the following problem: $$u_{tt}{(t,x)-(\alpha +\beta \parallel \nabla u(t,x)\parallel }_2^2+{\beta \parallel \nabla v(t,x)\parallel }_2^2)\Delta u(t,x)+\delta |u_t{(t,x)|}^{p-1}{u}_t(t,x)={\mu \left|u(t,x)\right|}^{q-1}u(t,x),x\in \Omega ,t\ge 0,v_{tt}{(t,x)-(\alpha +\beta \parallel \nabla u(t,x)\parallel }_2^2+{\beta \parallel \nabla v(t,x)\parallel }_2^2)\Delta v(t,x)+\delta |v_t{(t,x)|}^{p-1}{v}_t(t,x)={\mu \left|v(t,x)\right|}^{q-1}v(t,x),x\in \Omega ,t\ge 0,u(0,x)=u_0\left(x\right),u_t(0,x)=u_1\left(x\right),x\in \Omega ,v(0,x)=v_0\left(x\right),v_t(0,x)=v_1\left(x\right),x\in \Omega ,u{{|}_{{}_{\partial \Omega }}=v|}_{{}_{\partial \Omega }}=0$$ where $q>1$, $p\ge 1$, $\delta >0$, $\alpha >0$, $\beta \ge 0$, $\mu \in ℝ$ and $\Delta$ is the Laplacian in ${ℝ}^N$.

We consider the homogeneous Schrödinger equation with a long-range potential and show that its solutions satisfying some a priori bound at infinity can asymptotically be expressed as a sum of incoming and outgoing distorted spherical waves. Coefficients of these waves are related by the scattering matrix. This generalizes a similar result obtained earlier in the short-range case.