Convergence to separate variables solutions for a degenerate parabolic equation with gradient source

The large time behaviour of nonnegative solutions to a quasilinear degenerate diffusion equation with a source term depending solely on the gradient is investigated. After a suitable rescaling of time, convergence to a unique profile is shown for global solutions. The proof relies on the half-relaxed limits technique within the theory of viscosity solutions and on the construction of suitable supersolutions and barrier functions to obtain optimal temporal decay rates and boundary estimates. Blowup of weak solutions is also studied

Multiscale Problems in Solidification Processes

Our objective is to describe solidification phenomena in alloy systems. In the classical approach, balance equations in the phases are coupled to conditions on the phase boundaries which are modelled as moving hypersurfaces. The Gibbs-Thomson condition ensures that the evolution is consistent with thermodynamics. We present a derivation of that condition by defining the motion via a localized gradient flow of the entropy. Another general framework for modelling solidification of alloys with multiple phases and components is based on the phase field approach. The phase boundary motion is then given by a system of Allen-Cahn type equations for order parameters. In the sharp interface limit, i.e., if the smallest length scale ± related to the thickness of the diffuse phase boundaries converges to zero, a model with moving boundaries is recovered. In the case of two phases it can even be shown that the approximation of the sharp interface model by the phase field model is of second order in ±. Nowadays it is not possible to simulate the microstructure evolution in a whole workpiece. We present a two-scale model derived by homogenization methods including a mathematical justification by an estimate of the model error

New Lepidoptera-Parasitoid Associations in Weedy Corn Plantings: A Potential Alternate Host for \u3ci\u3eOstrinia Nubilalis\u3c/i\u3e (Lepidoptera: Pyralidae) Parasitoids

Larvae of the common sooty wing, Pholisora catullus, and pupae of the yellow-collared scape moth, Cisseps Fulvicollis, were collected in corn plantings containing different manipulated, indigenous weed communities to determine if these Lepidoptera had parasitoid species in common with the European corn borer, Ostrinia nubilalis. Pholisora catullus larvae were collected from lambsquarter, Chenopodium album, and redroot pigweed, Amaranthus retroflexus, whereas pupae of C. Julvicollis were obtained from corn. Four parasitoid species were reared from P. catulIus: Cotesia pholisorae, Oncophanes americanu (Hymenoptera: Braconidae), Gambrus ultimus, and Sinophorus albipalpus (Hymenoptera: Ichneumonidae). Of these, O. americanus and S. albipalpus represent new host records. Gambrus ultimus, however, was probably parasitizing a primary parasitoid of P. catullus. Itoplectis conquisitor and Vulgichneumon brevicinctor (Hymenoptera: Ichneumonidae) were reared from C. fulvicollis; V. brevicinctor had not previously been associated with this host. Both species reared from C. fulvicollis and Gambrus ultimus have been reported from O. nubilalis

Convergence to steady states for radially symmetric solutions to a quasilinear degenerate diffusive Hamilton-Jacobi equation

Convergence to a single steady state is shown for non-negative and radially symmetric solutions to a diffusive Hamilton-Jacobi equation with homogeneous Dirichlet boundary conditions, the diffusion being the $p$-Laplacian operator, $p\ge 2$, and the source term a power of the norm of the gradient of $u$. As a first step, the radially symmetric and non-increasing stationary solutions are characterized

A finite element method for a fourth order surface equation with application to the onset of cell blebbing

A variational problem for a fourth order parabolic surface partial differential equation is discussed. It contains nonlinear lower order terms, on which we only make abstract assumptions, and which need to be defined for specified problems.We derive a semi-discrete scheme based on the surface finite element method, show a-priori error estimates, and use the analytical results to prove well-posedness. Furthermore, we present a computational framework where specific problems can be conveniently implemented and, later on, altered with relative ease. It uses a domain specific language implemented in Python. The high level program control can also be done within the Python scripting environment. The computationally expensive step of evolving the solution over time is carried out by binding to an efficient C++ software back-end. The study is motivated by cell blebbing, which can be instrumental for cell migration. Starting with a force balance for the cell membrane, we derive a continuum model for some mechanical and geometrical aspects of the onset of blebbing in a form that fits into the abstract framework. It is flexible in that it allows for amending force contributions related to membrane tension or the presence of linker molecules between membrane and cell cortex. Cell membrane geometries given in terms of a parametrisation or obtained from image data can be accounted for by the software. The use of a domain specific language to describe the model makes is straightforward to add additional effects such as reaction-diffusion equations modelling some biochemistry on the cell membrane.Some numerical simulations illustrate the approach

Analysis of the discontinuous Galerkin method for elliptic problems on surfaces

We extend the discontinuous Galerkin (DG) framework to a linear second-order elliptic problem on a compact smooth connected and oriented surface. An interior penalty (IP) method is introduced on a discrete surface and we derive a-priori error estimates by relating the latter to the original surface via the lift introduced in Dziuk (1988). The estimates suggest that the geometric error terms arising from the surface discretisation do not affect the overall convergence rate of the IP method when using linear ansatz functions. This is then verified numerically for a number of test problems. An intricate issue is the approximation of the surface conormal required in the IP formulation, choices of which are investigated numerically. Furthermore, we present a generic implementation of test problems on surfaces.Comment: 21 pages, 4 figures. IMA Journal of Numerical Analysis 2013, Link to publication: http://imajna.oxfordjournals.org/cgi/content/abstract/drs033? ijkey=45b23qZl5oJslZQ&keytype=re

The Bindweed Plume Moth, \u3ci\u3eEmmelina Monodactyla\u3c/i\u3e (Lepidoptera: Pterophoridae): First Host Record for \u3ci\u3ePhaeogenes Vincibilis\u3c/i\u3e (Hymenoptera: Ichneumonidae)

The first host record for Phaeogenes (= Oronotus) vincibilis, a solitary ichneumonine pupal parasite, is the bindweed plume moth, Emmelina monodactyla