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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: 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: 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 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: 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]