We consider a one-dimensional model biodegradation system consisting of two reactionadvection equations for nutrient and pollutant concentrations and a rate equation for biomass. The hydrodynamic dispersion is ignored. Under an explicit condition on the decay and growth rates of biomass, the system can be approximated by two component models by setting biomass kinetics to equilibrium. We derive closed form solutions for constant speed traveling fronts for the reduced two component models and compare their profiles in homogeneous media. For a spatially random velocity field, we introduce travel time and study statistics of degradation fronts via representations in terms of the travel time probability density function (<i>pdf</i>) and the traveling front profiles. The travel time <i>pdf</i> does not vary with the nutrient and pollutant concentrations and only depends on the random water velocity. The traveling front profiles

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We study the spectral geometric properties of the scalar Laplace–Beltrami operator associated to the Weil–Petersson metric $g_{WP}$ on $\mathcal{M}_{\gamma}$ , the Riemann moduli space of surfaces of genus  $\gamma>1$. This space has a singular compactification with respect to $g_{WP}$, and this metric has crossing cusp-edge singularities along a finite collection of simple normal crossing divisors. We prove first that the scalar Laplacian is essentially self-adjoint, which then implies that its spectrum is discrete.

We look for bounded periodic solutions for a parametric fractional problem involving a continuous nonlinearity with subcritical growth. By using a variant of Caffarelli and Silvestre extension method adapted to the periodic case and variational tools we prove the existence of at least three bounded periodic solutions when the parameter varies in an appropriate range.

We prove the nonlinear asymptotic stability of the gravitational hydrostatic equilibrium for the general equation of state of pressure-density relation in the framework of vacuum free boundary problem of spherically symmetric compressible Navier-Stokes-Poisson equations in three dimensions.The results apply to white dwarfs and polytropes with γ≥2, for which even the linearized asymptotic stability results are not available in literature, to the best of the authors' knowledge. Detailed decay rates of perturbations are given.