Existence and Uniqueness of Solutions to a Nonlocal Equation with Monostable Nonlinearity

Let $J \in C(\mathbb{R})$, $J\ge 0$, \int_{\tiny\mathbb{R}} J = 1 and consider the nonlocal diffusion operator $\mathcal{M}[u] = J \star u - u$. We study the equation $\mathcal{M} u + f(x,u) = 0$, $u \ge 0$, in $\mathbb{R}$, where $f$ is a KPP-type nonlinearity, periodic in $x$. We show that the principal eigenvalue of the linearization around zero is well defined and that a nontrivial solution of the nonlinear problem exists if and only if this eigenvalue is negative. We prove that if, additionally, $J$ is symmetric, then the nontrivial solution is unique

Global exponential convergence to variational traveling waves in cylinders

We prove, under generic assumptions, that the special variational traveling wave that minimizes the exponentially weighted Ginzburg-Landau functional associated with scalar reaction-diffusion equations in infinite cylinders is the long-time attractor for the solutions of the initial value problems with front-like initial data. The convergence to this traveling wave is exponentially fast. The obtained result is mainly a consequence of the gradient flow structure of the considered equation in the exponentially weighted spaces and does not depend on the precise details of the problem. It strengthens our earlier generic propagation and selection result for "pushed" fronts.Comment: 23 page

Endomorphisms of quantized Weyl algebras

Belov-Kanel and Kontsevich conjectured that the group of automorphisms of the n'th Weyl algebra and the group of polynomial symplectomorphisms of C^2 are canonically isomorphic. We discuss how this conjecture can be approached by means of (second) quantized Weyl algebras at roots of unity

Quenching and Propagation of Combustion Without Ignition Temperature Cutoff

We study a reaction-diffusion equation in the cylinder $\Omega = \mathbb{R}\times\mathbb{T}^m$, with combustion-type reaction term without ignition temperature cutoff, and in the presence of a periodic flow. We show that if the reaction function decays as a power of $T$ larger than three as $T\to 0$ and the initial datum is small, then the flame is extinguished -- the solution quenches. If, on the other hand, the power of decay is smaller than three or initial datum is large, then quenching does not happen, and the burning region spreads linearly in time. This extends results of Aronson-Weinberger for the no-flow case. We also consider shear flows with large amplitude and show that if the reaction power-law decay is larger than three and the flow has only small plateaux (connected domains where it is constant), then any compactly supported initial datum is quenched when the flow amplitude is large enough (which is not true if the power is smaller than three or in the presence of a large plateau). This extends results of Constantin-Kiselev-Ryzhik for combustion with ignition temperature cutoff. Our work carries over to the case $\Omega = \mathbb{R}^n\times\mathbb{T}^m$, when the critical power is $1 + 2/n$, as well as to certain non-periodic flows

Metallic liquid hydrogen and likely Al2O3 metallic glass

Dynamic compression has been used to synthesize liquid metallic hydrogen at 140 GPa (1.4 million bar) and experimental data and theory predict Al2O3 might be a metallic glass at ~300 GPa. The mechanism of metallization in both cases is probably a Mott-like transition. The strength of sapphire causes shock dissipation to be split differently in the strong solid and soft fluid. Once the 4.5-eV H-H and Al-O bonds are broken at sufficiently high pressures in liquid H2 and in sapphire (single-crystal Al2O3), electrons are delocalized, which leads to formation of energy bands in fluid H and probably in amorphous Al2O3. The high strength of sapphire causes shock dissipation to be absorbed primarily in entropy up to ~400 GPa, which also causes the 300-K isotherm and Hugoniot to be virtually coincident in this pressure range. Above ~400 GPa shock dissipation must go primarily into temperature, which is observed experimentally as a rapid increase in shock pressure above ~400 GPa. The metallization of glassy Al2O3, if verified, is expected to be general in strong oxide insulators. Implications for Super Earths are discussed.Comment: 8 pages, 5 figures, 14th Liquid and Amorphous Metals Conference, Rome 201

Entropy-Dominated Dissipation in Sapphire Shock-Compressed up to 400 GPa (4 Mbar)

Sapphire (single-crystal Al2O3) is a representative Earth material and is used as a window and/or anvil in shock experiments. Pressure, for example, at the core-mantle boundary is about 130 gigapascals (GPa). Defects induced by 100-GPa shock waves cause sapphire to become opaque, which precludes measuring temperature with thermal radiance. We have measured wave profiles of sapphire crystals with several crystallographic orientations at shock pressures of 16, 23, and 86 GPa. At 23 GPa plastic-shock rise times are generally quite long (~100 ns) and their values depend sensitively on the direction of shock propagation in the crystal lattice. The long rise times are probably caused by the high strength of inter-atomic interactions in the ordered three-dimensional sapphire lattice. Our wave profiles and recent theoretical and laser-driven experimental results imply that sapphire disorders without significant shock heating up to about 400 GPa, above which Al2O3 is amorphous and must heat. This picture suggests that the characteristic shape of shock compression curves of many Earth materials at 100 GPa pressures is caused by a combination of entropy and temperature.Comment: 12 pages, 4 figure

When the heart kills the liver: acute liver failure in congestive heart failure

Congestive heart failure as a cause of acute liver failure is rarely documented with only a few cases