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In this paper, we continue to study a spatially heterogeneous predator–prey system where the interaction is governed by a Holling type II functional response, which has been studied in Du and Shi (2007) [14] . We further study the asymptotic profile of positive solutions and give a complete understanding of coexistence region. Moreover, a good understanding of the number, stability and asymptotic behavior of positive solutions is gained for large m . Finally, we further compare the difference of steady-state solutions between m > 0 and m = 0 . It turns out that the spatial heterogeneity of the environment and the Holling type II functional response play a very important role in this model. [ABSTRACT FROM AUTHOR]

In this paper, we provide two-sided estimates and uniform asymptotics for the solution of d -dimensional critical fractal Burgers equation u t − Δ α / 2 u + b ⋅ ∇ ( u | u | q ) = 0 , α ∈ ( 1 , 2 ) , b ∈ R d for q = ( α − 1 ) / d and u 0 ∈ L 1 ( R d ) . We consider also q > ( α − 1 ) / d under additional condition u 0 ∈ L ∞ ( R d ) . In both cases we assume u 0 ≥ 0 , which implies that the solution is non-negative. The estimates are given in the terms of the function P t u 0 , where P t denotes the semigroup for the operator ∂ t − Δ α / 2 . [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we consider two classes of free boundary value problems of a viscous two-phase liquid-gas model relevant to the flow in wells and pipelines with mass-dependent viscosity coefficient. The liquid is treated as an incompressible fluid whereas the gas is assumed to be polytropic. We obtain the asymptotic behavior and decay rates of the mass functions , when the initial masses are assumed to be connected to vacuum both discontinuously and continuously, which improves the corresponding result about Navier–Stokes equations in Zhu (2010) . [Copyright &y& Elsevier]