Drinking Bourbon with Cupid

It was Valentine’s Day, and rather than enjoying the suspiciously commercial holiday with a romantic partner, I was alone watching reruns of “How I Met Your Mother,” from a cozy armchair with a cigar in one hand and a glass of bourbon in the other. The show prompted me to examine the nature of relationships; specifically, how the media portrays them vastly different than reality and the implications that arise as a result. Romantic relationships in film and literature appear to be idealized to a ridiculous degree. Unfortunately for us, this means that we create unrealistic expectations for our partners that lead many to remain single while they search for a relationship that adheres to the media’s extravagant standards. [excerpt

Attractors for Damped Semilinear Wave Equations with Singularly Perturbed Acoustic Boundary Conditions

Under consideration is the damped semilinear wave equation $$u_{tt}+u_t-\Delta u+u+f(u)=0$$ in a bounded domain $\Omega$ in $\mathbb{R}^3$ subject to an acoustic boundary condition with a singular perturbation, which we term "massless acoustic perturbation," \ep\delta_{tt}+\delta_t+\delta = -u_t\quad\text{for}\quad \ep\in[0,1]. By adapting earlier work by S. Frigeri, we prove the existence of a family of global attractors for each \ep\in[0,1]. We also establish the optimal regularity for the global attractors, as well as the existence of an exponential attractor, for each \ep\in[0,1]. The later result insures the global attractors possess finite (fractal) dimension, however, we cannot yet guarantee that this dimension is independent of the perturbation parameter \ep. The family of global attractors are upper-semicontinuous with respect to the perturbation parameter \ep, a result which follows by an application of a new abstract result also contained in this article. Finally, we show that it is possible to obtain the global attractors using weaker assumptions on the nonlinear term $f$, however, in that case, the optimal regularity, the finite dimensionality, and the upper-semicontinuity of the global attractors does not necessarily hold.Comment: To appear in EJDE. arXiv admin note: substantial text overlap with arXiv:1503.01821 and text overlap with arXiv:1302.426

Global existence of weak solutions for strongly damped wave equations with nonlinear boundary conditions and balanced potentials

We demonstrate the global existence of weak solutions to a class of semilinear strongly damped wave equations possessing nonlinear hyperbolic dynamic boundary conditions. Our work assumes $(-\Delta_W)^\theta \partial_tu$ with $\theta\in[\frac{1}{2},1)$ and where $\Delta_W$ is the Wentzell-Laplacian. Hence, the associated linear operator admits a compact resolvent. A balance condition is assumed to hold between the nonlinearity defined on the interior of the domain and the nonlinearity on the boundary. This allows for arbitrary (supercritical) polynomial growth on each potential, as well as mixed dissipative/anti-dissipative behavior. Moreover, the nonlinear function defined on the interior of the domain is assumed to be only $C^0$

Infrared Imaging of Planetary Nebulae from the Ground Up

New ground-based telescopes and instruments, the return of the NICMOS instrument on the Hubble Space Telescope (HST), and the recent launch of the Spitzer Space Telescope have provided new tools that are being utilized in the study of planetary nebulae. Multiwavelength, high spatial resolution ground-based and HST imaging have been used to probe the inner regions of young PNe to determine their structure and evaluate formation mechanisms. Spitzer/IRAC and MIPS have been used to image more evolved PNe to determine the spatial distribution of molecular hydrogen, ionized gas, and dust in the nebulae and halos.Comment: 8 pages, 3 figures, invited review given at IAU Symp. 234, to appear in "Planetary Nebulae in Our Galaxy and Beyond", eds. M. J. Barlow & R. H. Mende