By analysing the uniform attractor for multi-valued processes, we study the long-time behaviour of the solutions of a model of non-autonomous porous-medium equations. The result is obtained by using the <i>a priori</i> estimates and suitable compactness arguments.

We consider a family \{\mathcal {H}^arepsilon\} _ {arepsilon&gt; 0} of \{\mathcal {H}^arepsilon\} _ {arepsilon&gt; 0} -periodic Schrdinger operators with \{\mathcal {H}^arepsilon\} _ {arepsilon&gt; 0} -interactions supported on a lattice of closed compact surfaces; within a minimal period cell one has \{\mathcal {H}^arepsilon\} _ {arepsilon&gt; 0} surfaces. We show that in the limit when \{\mathcal {H}^arepsilon\} _ {arepsilon&gt; 0} and the interactions strengths are appropriately scaled, \{\mathcal {H}^arepsilon\} _ {arepsilon&gt; 0} has at most \{\mathcal {H}^arepsilon\} _ {arepsilon&gt; 0} gaps within finite intervals, and moreover, the limiting behavior of the first \{\mathcal {H}^arepsilon\} _ {arepsilon&gt; 0} gaps can be completely controlled through a suitable choice of those surfaces and of the interactions strengths.

In this paper, we consider the three-dimensional Schrdinger operator with a -interaction of strength &gt; 0 supported on an unbounded surface parametrized by the mapping R2x(x,f(x)), where 0, and f:R2R, f 0, is a C2-smooth, compactly supported function. The surface supporting the interaction can be viewed as a local deformation of the plane. It is known that the essential spectrum of this Schrdinger operator coincides with 142,+. We prove that for all sufficiently small &gt; 0, its discrete spectrum is non-empty and consists of a unique simple eigenvalue. Moreover, we obtain an asymptotic expansion of this eigenvalue in the limit 0+. In particular, this eigenvalue tends to 142 exponentially fast as 0+.

We consider the self-adjoint Smilansky Hamiltonian H in L2 R2) associated with the formal differential expression -x2-1/2(y2+y2)-2y(x) in the sub-critical regime, (0, 1). We demonstrate the existence of resonances for H on a countable subfamily of sheets of the underlying Riemann surface whose distance from the physical sheet is finite. On such sheets, we find resonance free regions and characterize resonances for small &gt; 0. In addition, we refine the previously known results on the bound states of H in the weak coupling regime (0+). In the proofs we use Birman-Schwinger principle for H, elements of spectral theory for Jacobi matrices, and the analytic implicit function theorem.

It is known that the maximum diameter for the rupture-risk assessment of the abdominal aortic aneurysm is a generally good method, but not sufficient. Alternative features obtained with computational modeling may provide additional useful criteria. Though computational approaches are noninvasive, they are often time-consuming because of the high computational complexity. In this paper, we present a highly parallel algorithm for the numerical simulation of unsteady blood flows in the patient-specific abdominal aorta. We model the blood flow with the unsteady incompressible Navier-Stokes equations, and solve the discretized system with a highly scalable domain decomposition method. With this approach, the complete flow field can be obtained in less than an hour, instead of days with older methods. We show experimentally that the proposed method offers accurate solutions of the pressure, the velocity and the wall shear stress, and the parallel efficiency is higher than 70% on a parallel computer with more than 1,000 processor cores.