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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]

This paper concerns the equation (0.1) u t = u x x + f ( x − c t , u ) , x ∈ R , where c ≥ 0 is a forcing speed and f : ( s , u ) ∈ R × R + → R is asymptotically of KPP type as s → − ∞ . We are interested in the questions of whether such a forced moving KPP nonlinearity from behind can give rise to traveling waves with the same speed and how they attract solutions of initial value problems when they exist. Under a sublinearity condition on f ( s , u ) , we obtain the complete existence and multiplicity of forced traveling waves as well as their attractivity except for some critical cases. In these cases, we provide examples to show that there is no definite answer unless one imposes further conditions depending on the heterogeneity of f in s ∈ R . [ABSTRACT FROM AUTHOR]

This paper deals with the following linearly coupled nonlinear Kirchhoff-type system: { − ( a 1 + b 1 ∫ R 3 | ∇ u | 2 d x ) Δ u + μ 1 u = f ( u ) + β v in R 3 , − ( a 2 + b 2 ∫ R 3 | ∇ v | 2 d x ) Δ v + μ 2 v = g ( v ) + β u in R 3 , u , v ∈ H 1 ( R 3 ) , where a i > 0 , b i ≥ 0 , μ i > 0 are constants for i = 1 , 2 , β > 0 is a parameter and f , g ∈ C ( R , R ) . Under the general Berestycki–Lions type assumptions on f and g , we establish the existence of positive vector solutions and positive vector ground state solutions respectively by using variational methods. We also study the asymptotic behavior of these solutions as β → 0 + . [ABSTRACT FROM AUTHOR]

Abstract: The purpose of this paper is to investigate the relation between the moments and the asymptotic behavior of solutions to the Burgers equation. The Burgers equation is a special nonlinear problem that turns into a linear one after the Cole–Hopf transformation. Our asymptotic analysis depends on this transformation. In this paper an asymptotic approximate solution is constructed, which is given by the inverse Cole–Hopf transformation of a summation of n heat kernels. The k-th order moments of the exact and the approximate solution are contracting with order in -norm as . This asymptotics indicates that the convergence order is increased by a similarity scale whenever the order of controlled moments is increased by one. The theoretical asymptotic convergence orders are tested numerically. [Copyright &y& Elsevier]

Abstract: In this paper, we study a class of semilinear functional evolution equations in which the nonlinearity is demicontinuous and satisfies a semimonotone condition. We prove the existence, uniqueness and exponentially asymptotic stability of the mild solutions. Our approach is to apply a convenient version of Burkholder inequality for convolution integrals and an iteration method based on the existence and measurability results for the functional integral equations in Hilbert spaces. An Itô-type inequality is the main tool to study the uniqueness, p-th moment and almost sure sample path asymptotic stability of the mild solutions. We also give some examples to illustrate the applications of the theorems and meanwhile we compare the results obtained in this paper with some others appeared in the literature. [Copyright &y& Elsevier]

This paper deals with a free boundary problem for diffusion equation with a certain class of bistable nonlinearity which allows two positive stable equilibrium states as an ODE model. This problem models the invasion of a biological species and the free boundary represents the spreading front of its habitat. Our main interest is to study large-time behaviors of solutions for the free boundary problem. We will completely classify asymptotic behaviors of solutions and, in particular, observe two different types of spreading phenomena corresponding to two positive stable equilibrium states. Moreover, it will be proved that, if the free boundary expands to infinity, an asymptotic speed of the moving free boundary for large time can be uniquely determined from the related semi-wave problem. [ABSTRACT FROM AUTHOR]

Abstract: This paper is concerned with asymptotic profiles of ground states for a class of quasilinear Schrödinger equations with -supercritical nonlinearities. We study the blow-up rate of ground states by using the dual variational structure of equations as well as various variational methods. [Copyright &y& Elsevier]

Abstract: In this paper, we establish the existence and the nonlinear stability of traveling wave solutions to a system of conservation laws which is transformed, by a change of variable, from the well-known Keller–Segel model describing cell (bacteria) movement toward the concentration gradient of the chemical that is consumed by the cells. We prove the existence of traveling fronts by the phase plane analysis and show the asymptotic nonlinear stability of traveling wave solutions without the smallness assumption on the wave strengths by the method of energy estimates. [Copyright &y& Elsevier]

Abstract: We study semilinear elliptic equations in a generally unbounded domain when the pertinent quadratic form is nonnegative and the potential is generally singular, typically a homogeneous function of degree −2. We prove solvability results based on the asymptotic behavior of the potential with respect to unbounded translations and dilations, while the nonlinearity is a perturbation of a self-similar, possibly oscillating, term of critical growth satisfying , , . This paper focuses on two qualitatively different cases of this problem, one when the quadratic form has a generalized ground state and another where the presence of potential does not change the energy space. In the latter case we allow nonlinearities with oscillatory critical growth. An important example of such quadratic form is the one on with the radial Hardy potential with in the first case, in the second case, where is the largest constant for which the energy form remains nonnegative. [Copyright &y& Elsevier]

Abstract: In this paper, we study exterior Neumann problems with an asymptotically linear nonlinearity. We establish the existence of ground state solutions. Furthermore when the domain is a complement of a ball, we prove the ground state solutions are not radially symmetric. We also give asymptotic profiles of ground state solutions. [Copyright &y& Elsevier]