We obtain the existence and uniqueness of solutions for a class of retarded parabolic equations with superlinear perturbations. The asymptotic behavior result is studied by using the pullback attractor framework.

G-equations are well-known front propagation models in turbulent combustion which describe the front motion law in the form of local normal velocity equal to a constant (laminar speed) plus the normal projection of fluid velocity. In level set formulation, G-equations are HamiltonJacobi equations with convex (<i>L</i> ¹ type) but non-coercive Hamiltonians. Viscous G-equations arise from either numerical approximations or regularizations by small diffusion. The nonlinear eigenvalue $${\bar H}$$ from the cell problem of the viscous G-equation can be viewed as an approximation of the inviscid turbulent flame speed <i>s</i> _T. An important problem in turbulent combustion theory is to study properties of <i>s</i> _T, in particular how <i>s</i> _T depends on the flow amplitude <i>A</i>. In this paper, we study the behavior of $${\bar H=\bar H(A,d)}$$ as <i>A</i> + at any fixed diffusion constant <i>d</i> &gt; 0. For cellular flow, we show that$$\bar H(A,d)\leqq

Speed ensemble of bistable (combustion) fronts in mean zero stationary Gaussian shear flows inside two and three dimensional channels is studied with a min-max variational principle. In the small root mean square regime of shear flows, a new class of multi-scale test functions are found to yield speed asymptotics. The quadratic speed enhancement law holds with probability arbitrarily close to one under the almost sure continuity (dimension two) and mean square Hlder regularity (dimension three) of the shear flows. Remarks are made on the quadratic and linear laws of front speed expectation in the small and large root mean square regimes.

ABSTRACT We study coagulation-fragmentation equations inspired by a simple model proposed in fisheries science to explain data for the size distribution of schools of pelagic fish. Although the equations lack detailed balance and admit no $H$-theorem, we are able to develop a rather complete description of equilibrium profiles and large-time behavior, based on recent developments in complex function theory for Bernstein and Pick functions. In the large-population continuum limit, a scaling-invariant regime is reached in which all equilibria are determined by a single scaling profile. This universal profile exhibits power-law behavior crossing over from exponent $-\frac23$ for small size to $-\frac32$ for large size, with an exponential cut-off.

In the present paper, we consider the initial value problem for the bipolar Vlasov-Poisson-Boltzmann (bVPB) system and its corresponding modified Vlasov-Poisson-Boltzmann (mVPB). We give the spectrum analysis on the linearized bVPB and mVPB systems around their equilibrium state and show the optimal convergence rate of global solutions. It was showed that the electric field decays exponentially and the distribution function tends to the absolute Maxwellian at the optimal convergence rate (1+ t)^{-3/4} for the bVPB system, yet both the electric field and the distribution function converge to equilibrium state at the optimal rate (1+ t)^{-3/4} for the mVPB system.