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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 investigate the global existence and uniqueness of strong solutions to the initial boundary value problem for the 3D compressible Navier–Stokes equations without heat conductivity in a bounded domain with slip boundary. The global existence and uniqueness of strong solutions are obtained when the initial data is near its equilibrium in H 2 ( Ω ) . Furthermore, the exponential convergence rates of the pressure and velocity are also proved by delicate energy methods. [ABSTRACT FROM AUTHOR]

The goal of this paper is to study the behavior of certain solutions to the Swift–Hohenberg equation on a one-dimensional torus T . Combining results from Γ-convergence and ODE theory, it is shown that solutions corresponding to initial data that is L 1 -close to a jump function v , remain close to v for large time. This can be achieved by regarding the equation as the L 2 -gradient flow of a second order energy functional, and obtaining asymptotic lower bounds on this energy in terms of the number of jumps of v . [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: This paper deals with the existence and stability of periodic pyramidal traveling fronts for reaction–diffusion equations with bistable time-periodic nonlinearity in with . It is well known that two-dimensional periodic traveling curved fronts exist and are stable. In this paper, by constructing various of supersolutions and subsolutions, we first show that there exist three-dimensional periodic pyramidal traveling fronts, and then we prove that such periodic pyramidal traveling fronts are asymptotically stable. Finally, we further prove that our existence result holds for with . [Copyright &y& Elsevier]

Abstract: This paper deals with a reinforcement problem for a plane domain whose boundary is a deterministic or random “mixture” of self-similar Koch curves. We construct an ε-thin polygonal 2-dimensional fiber , , , around pre-fractal approximating domains and related suitable energy functionals. The aim of this paper is to study the asymptotic behavior of the reinforced energy functionals while, simultaneously, the thickness of the fibers and the conductivity of the functionals on the fibers converges to 0 as . [Copyright &y& Elsevier]

Abstract evolution differential equations of second order in time are studied in order to get (almost) optimal decay estimates to the corresponding energy functional of the equations. The framework is supported by a special energy method in the associated Fourier space. The constructed abstract theory can be applied to several concrete evolutionary partial differential equations as is illustrated in the last section of the paper. [ABSTRACT FROM AUTHOR]

In this paper, we apply level set and nonlinear perturbation methods to obtain the asymptotic behavior of the solution to a kind of parabolic Monge–Ampère equation at infinity. The Jörgens–Calabi–Pogorelov theorem for parabolic and elliptic Monge–Ampère equation can be regarded as special cases of our result. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we show the existence of a positive solution for the following class of elliptic equations where f is a continuous function with quasicritical growth and K is a nonnegative function verifying some conditions of two kinds. We analyze the problem with K asymptotically periodic, K periodic and K belonging to some . [Copyright &y& Elsevier]

Abstract: In this paper, the -index is defined and the -boundary value problem of Hamiltonian system is studied. As applications, the existence and multiplicity results of periodic solutions of asymptotically linear delay differential systems and delay Hamiltonian systems are obtained. [Copyright &y& Elsevier]

Abstract: In this paper, we use the potential analysis to study the properties of the positive solutions of a γ-Laplace system in Here , satisfy the critical condition . First, the positive solutions u and v satisfy an integral system involving the Wolff potentials. We then use the method of regularity lifting to obtain an optimal integrability for this Wolff type integral system. Different from the case of , it is more difficult to handle the asymptotics since u and v have not radial structures. We overcome this difficulty by a new method and obtain the decay rates of u and v as . We believe that this new method is appropriate to deal with the asymptotics of other decaying solutions without the radial structures. [Copyright &y& Elsevier]

Abstract: In this paper we consider the set of equations describing Oldroyd-B fluids in exterior domains. It is shown that these equations admit a unique, global solution defined in a certain function space provided the initial data and the coupling constant are small enough. [Copyright &y& Elsevier]

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]

Abstract: This paper is concerned with the large time behavior of solutions to a radiating gas model, which is represented mathematically as a Cauchy problem for a one-dimensional hyperbolic–elliptic coupled system, with suitably given far field states. Suppose the corresponding Riemann problem for the Euler system admits a contact discontinuity wave, then we can construct a “viscous contact wave” for such a hyperbolic–elliptic system. Based on the energy methods and the ellipticity of the radiation flux equation, we prove that the “viscous contact wave” is asymptotically stable provided that the strength of contact discontinuity and the perturbation of the initial data are suitably small. [Copyright &y& Elsevier]

Abstract: In this paper, we study the existence of infinitely many periodic solutions for second order Hamiltonian systems , where is either asymptotically quadratic or superquadratic as . [Copyright &y& Elsevier]

Abstract: We study the singularly perturbed state-dependent delay-differential equation which is a special case of the equation One knows that for every sufficiently small , Eq. (⁎) possesses at least one so-called slowly oscillating periodic solution, and moreover, the graph of every such solution approaches a specific sawtooth-like shape as . In this paper we obtain higher-order asymptotics of the sawtooth, including the location of the minimum and maximum of the solution with the form of the solution near these turning points, and as well an asymptotic formula for the period. Using these and other asymptotic formulas, we further show that the solution enjoys the property of superstability, namely, the nontrivial characteristic multipliers are of size for small ε. This stability property implies that this solution is unique among all slowly oscillating periodic solutions, again for small ε. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the stationary problem of a prey–predator cross-diffusion system with a protection zone for the prey. We discuss the existence and non-existence of coexistence states of the two species by using the bifurcation theory. As a result, it is shown that the cross-diffusion for the prey has beneficial effects on the survival of the prey when the intrinsic growth rate of the predator is positive. We also study the asymptotic behavior of positive stationary solutions as the cross-diffusion coefficient of the prey tends to infinity. [Copyright &y& Elsevier]

Abstract: Classical conditions for asymptotic stability of periodic solutions bifurcating from a limit cycle rely on the sign of the derivative of the associated bifurcation function at a zero. In this paper we show that, for analytic systems, this result is of topological nature. This means that it is enough to impose a change of sign at the zero, without any assumption on the succesive derivatives. [Copyright &y& Elsevier]

Abstract: In this paper we investigate the growth/decay rate of solutions of an abstract integral equation which frequently arises in quasilinear differential equations applying a variation-of-constants formula. These results are applicable to some abstract equations which appear in the theory of age-dependent population models and also to some quasilinear delay differential equations with bounded and unbounded delays. Examples are given to illustrate the sharpness of the results. [ABSTRACT FROM AUTHOR]

Abstract: This paper studies the uniqueness and the asymptotic stability of a pyramidal traveling front in the three-dimensional whole space. For a given admissible pyramid we prove that a pyramidal traveling front is uniquely determined and that it is asymptotically stable under the condition that given perturbations decay at infinity. For this purpose we characterize the pyramidal traveling front as a combination of planar fronts on the lateral surfaces. Moreover we characterize the pyramidal traveling front in another way, that is, we write it as a combination of two-dimensional V-form waves on the edges. This characterization also uniquely determines a pyramidal traveling front. [Copyright &y& Elsevier]

Abstract: Under some nondegeneracy conditions we give asymptotic formulae for the stability parameter of a family of singular-limit Hill''s equation which depends on three parameters. We use the blow-up techniques introduced in [R. Martínez, A. Samà, C. Simó, Analysis of the stability of a family of singular-limit linear periodic systems in . Applications, J. Differential Equations 226 (2006) 652–686]. The main contribution of this paper concerns the study of the nondegeneracy conditions. We give a geometrical interpretation of them, in terms of heteroclinic orbits for some related systems. In this way one can determine values of the parameters such that the nondegeneracy conditions are satisfied. As a motivation and application we consider the vertical stability of homographic solutions in the three-body problem. [Copyright &y& Elsevier]

Abstract: In this paper, we study positive periodic solutions to the repulsive singular perturbations of the Hill equations. It is proved that such a perturbation problem has at least two positive periodic solutions when the anti-maximum principle holds for the Hill operator and the perturbation is superlinear at infinity. The proof relies on a nonlinear alternative of Leray–Schauder type and on Krasnoselskii fixed point theorem on compression and expansion of cones. [Copyright &y& Elsevier]

The propagation of surface water waves in a frictional channel with a uniformly inclined bed is governed by a two-dimensional shallow river model. In this paper, we consider the time-asymptotic stability of weak planar travelling waves for a two-dimensional shallow river model with Darcy''s law. We derive an effective parabolic equation to analyze the wave front motion. By employing weighted energy estimates, we show that weak planar travelling waves are time-asymptotically stable under sufficiently small perturbations. [Copyright &y& Elsevier]