We show that positive$α$-stable densities are hyperbolically completely monotone if and only if$α$≤1/2. This gives a positive answer to a question raised by L. Bondesson in 1977.

We prove the existence of multidimensional traveling-wave solutions to the scalar equation for the transport of solutes (contaminants) with nonlinear adsorption and spatially periodic convection-diffusion-adsorption coefficients under the assumption that the nonlinear adsorption function satisfies the Lax and Oleinik entropy conditions. In the nondegenerate case, we also prove the uniqueness of the traveling waves. These traveling waves are analogues of viscous shock profiles. They propagate with effective speeds that depend on the periodic porous media only up to their mean states, and are given by an averaged Rankine-Hugoniot relation. This is a direct consequence of the fact that the transport equation is in conservation form. We use the sliding domain method, the continuation method, spectral theory, maximum principles, and a priori estimates. In the degenerate case, the traveling waves are weak

Abstract We study a class of fourth order nonlinear parabolic equations which include the thin-film equation and the quantum drift-diffusion model as special cases. We investigate these equations by first developing functional inequalities of the type $$ \int_\Omega u^{2\gamma-\alpha-\beta}\Delta\ua\Delta\ub dx \geq c\int_\Omega|\Delta \ur |^2dx, $$ which seem to be of interest on their own right.

The nonlinear asymptotic stability of Lane-Emden solutions is proved in this paper for spherically symmetric motions of viscous gaseous stars with the density dependent shear and bulk viscosities which vanish at the vacuum, when the adiabatic exponent γ lies in the stability regime (4/3, 2), by establishing the global-in-time regularity uniformly up to the vacuum boundary for the vacuum free boundary problem of the compressible Navier-Stokes-Poisson systems with spherical symmetry, which ensures the global existence of strong solutions capturing the precise physical behavior that the sound speed is C1/2-Hölder continuous across the vacuum boundary, the large time asymptotic uniform convergence of the evolving vacuum boundary, density and velocity to those of Lane-Emden solutions with detailed convergence rates, and the detailed large time behavior of solutions near the vacuum boundary. Those uniform convergence are of fundamental importance in the study of vacuum free boundary problems which are missing in the previous results for global weak solutions. Moreover, the results obtained in this paper apply to much broader cases of viscosities than those in Fang and Zhang (Arch Ration Mech Anal 191:195–243, 2009) for the theory of weak solutions when the adiabatic exponent γ lies in the most physically relevant range. Finally, this paper extends the previous local-in-time theory for strong solutions to a global-in-time one.

The paper aims at analytically exhibiting for the first time the fundamental mechanisms underlying the fact that effective biological tissue electrical properties and their frequency dependence reflect the tissue composition and physiology. For doing so, a homogenization theory is derived to describe the effective admittivity of cell suspensions. A new formula is reported for dilute cases that gives the frequency-dependent effective admittivity with re- spect to the membrane polarization. Different microstructures are shown to be distinguish- able via spectroscopic measurements of the overall admittivity using the spectral properties of the membrane polarization. The Debye relaxation times associated with the membrane polarization tensor are shown to be able to give the microscopic structure of the medium. A natural measure of the admittivity anisotropy is introduced and its dependence on the frequency of applied current is derived. A Maxwell-Wagner-Fricke formula is given for concentric circular cells, and the results can be extended to the random cases. A randomly deformed periodic medium is also considered and a new formula is derived for the overall admittivity of a dilute suspension of randomly deformed cells.