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Abstract The goal of this paper is to develop the Equivariant Degree based method for studying relative periodic solutions in the setiings with lack of smoothness and/or genericity. In this paper, we consider an equivariant Hopf bifurcation of relative periodic solutions from relative equilibria in systems of functional differential equations respecting Γ × S 1 -spatial symmetries. The existence of branches of relative periodic solutions together with their symmetric classification is established using the equivariant twisted Γ × S 1 -degree with one free parameter. As a case study, we consider a delay differential model of coupled identical passively mode-locked semiconductor lasers with the dihedral symmetry group Γ = D 8 ; and, a system of hysterestic electro-mechanical oscillators coupled in the same symmetric fashion. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we consider an initial-boundary value problem of the p-Laplacian parabolic equation { u t (x , t) = div (| ∇ u (x , t) | p − 2 ∇ u (x , t)) + f (u (x , t)) , (x , t) ∈ Ω × (0 , + ∞) , u (x , t) = 0 , (x , t) ∈ ∂ Ω × [ 0 , + ∞) , u (x , 0) = u 0 ≥ 0 , x ∈ Ω ‾ , where p ≥ 2 and Ω is a bounded domain of R N (N ≥ 1) with smooth boundary ∂Ω. The main contribution of this work is to introduce a new condition (C p) α ∫ 0 u f (s) d s ≤ u f (u) + β u p + γ , u > 0 for some α , β , γ > 0 with 0 < β ≤ (α − p) λ 1 , p p , where λ 1 , p is the first eigenvalue of p-Laplacian Δ p , and we use the concavity method to obtain the blow-up solutions to the above equations. In fact, it will be seen that the condition (C p) improves the conditions ever known so far. [ABSTRACT FROM AUTHOR]

Abstract This paper is concerned with a time periodic competition–diffusion system { u t = u x x + u (r 1 (t) − a 1 (t) u − b 1 (t) v) , t > 0 , x ∈ R , v t = d v x x + v (r 2 (t) − a 2 (t) u − b 2 (t) v) , t > 0 , x ∈ R , where u (t , x) and v (t , x) denote the densities of two competing species, d > 0 is some constant, r i (t) , a i (t) and b i (t) are T -periodic continuous functions. Under suitable conditions, it has been confirmed by Bao and Wang (2013) [2] that this system admits periodic traveling fronts connecting two stable semi-trivial T -periodic solutions (p (t) , 0) and (0 , q (t)) associated to the corresponding kinetic system. In the present work, we first investigate the asymptotic behavior of periodic bistable traveling fronts with non-zero speeds at infinity by a dynamical approach combined with the two-sided Laplace transform method. With these asymptotic properties, we then obtain some key estimates. As a result, by applying the super- and subsolutions techniques as well as the comparison principle, we establish the existence and various qualitative properties of the so-called entire solutions defined for all time and the whole space, which provides some new spreading ways other than periodic traveling waves for two strongly competing species interacting in a heterogeneous environment. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we prove that the spectral radius of the traction double layer potential operator associated with the Lamé system on an infinite sector in is within 10−2 from a certain conjectured value which depends explicitly on the aperture of the sector and the Lamé moduli of the system. This type of result is relevant to the spectral radius conjecture, cf., e.g., Problem 3.2.12 in [C.E. Kenig, Harmonic Analysis Techniques for Second Order Elliptic Boundary Value Problems, CBMS Reg. Conf. Ser. Math., vol. 83, Amer. Math. Soc., Providence, RI, 1994]. The techniques employed in the paper are a blend of classical tools such as Mellin transforms, and Calderón–Zygmund theory, as well as interval analysis—resulting in a computer-aided proof. [Copyright &y& Elsevier]

Abstract: In this paper we study the period function of centers of planar polynomial differential systems. With a convenient compactification of the phase portrait, the boundary of the period annulus of the center has two connected components: the center itself and a polycycle. We are interested in the behaviour of the period function near the polycycle. The desingularization of its critical points gives rise to a new polycycle (monodromic as well) with hyperbolic saddles or saddle-nodes at the vertices. In this paper we compute the first terms in the asymptotic development of the time function around any orbitally linearizable saddle that may come from this desingularization process. In addition, we use these developments to study the bifurcation diagram of the period function of the dehomogenized Loud''s centers. More generally, the tools developed here can be used to study the return time function around a monodromic polycycle. This work is a continuation of the results in [P. Mardešić, D. Marín, J. Villadelprat, On the time function of the Dulac map for families of meromorphic vector fields, Nonlinearity 16 (2003) 855–881; P. Mardešić, D. Marín, J. Villadelprat, The period function of reversible quadratic centers, J. Differential Equations 224 (2006) 120–171]. [Copyright &y& Elsevier]

Abstract: In this paper we are interested in the global existence and large-time behavior of solutions to the initial-boundary value problem for subcritical Kuramoto–Sivashinsky-type equationwhere the nonlinear term depends on the unknown function u and its derivative and satisfy the estimatewith such that The aim of this paper is to prove the global existence of solutions to the initial-boundary value problem (0.1) in subcritical case, when the nonlinear term has a time decay rate less than that of the linear terms of Eq. (0.1). Also we find the main term of the asymptotic representation of solutions. [Copyright &y& Elsevier]

Abstract: Motivated by the open problems on the generic convergence of cooperative systems without the assumption of irreducibility independently proposed by Smith and Sontag, this paper investigates the generic convergence for the solutions of cooperative cascade systems with length one. First, by fixing a solution of a base system converging to an equilibrium, we establish both the Nonordering of Limit Sets and the Limit Set Dichotomy for the solutions of the cascade system. Combining these tools with the idea of limiting equation, we then prove the Sequential Limit Set Trichotomy and hence the quasiconvergence in generic meaning. The generic convergent result is finally obtained by improving the Limit Set Dichotomy. [Copyright &y& Elsevier]

Abstract: We address the analysis of a nonlinear and degenerating PDE system, proposed by M. Frémond for modelling phase transitions in viscoelastic materials subject to thermal effects. The system features an internal energy balance equation, governing the evolution of the absolute temperature ϑ, an evolution equation for the phase change parameter χ, and a stress–strain relation for the displacement variable u. The main novelty of the model is that the equations for χ and u are coupled in such a way as to take into account the fact that the properties of the viscous and of the elastic parts influence the phase transition phenomenon in different ways. However, this brings about an elliptic degeneracy in the equation for u which needs to be carefully handled. In this paper, we first prove a local (in time) well-posedness result for (a suitable initial–boundary value problem for) the above mentioned PDE system, in the (spatially) three-dimensional setting. Secondly, we restrict to the one-dimensional case, in which, for the same initial–boundary value problem, we indeed obtain a global well-posedness theorem. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the initial–boundary value problem of the generalized Benjamin–Bona–Mahony–Burgers equation in the half-space Here is an unknown function of and , are two given constant states and the nonlinear function is assumed to be a strictly convex function of u. We first show that the corresponding boundary layer solution of the above initial–boundary value problem is global nonlinear stable and then, by employing the space–time weighted energy method which was initiated by Kawashima and Matsumura [S. Kawashima, A. Matsumura, Asymptotic stability of travelling wave solutions of systems for one-dimensional gas motion, Comm. Math. Phys. 101 (1985) 97–127], the convergence rates (both algebraic and exponential) of the global solution to the above initial–boundary value problem toward the boundary layer solution are also obtained for both the non-degenerate case and the degenerate case . [Copyright &y& Elsevier]

Abstract: In this paper, we establish the global attractivity of the positive steady state of the diffusive Nicholson''s equation with homogeneous Neumann boundary value under a condition that makes the equation a non-monotone dynamical system. To achieve this, we develop a novel method: combining a dynamical systems argument with maximum principle and some subtle inequalities. [Copyright &y& Elsevier]

Abstract: This paper examines the time-dependent double diffusive convection in Brinkman flow in a semi-infinite cylinder. Under appropriate initial and boundary conditions the authors establish exponential decay of solutions in “energy” norm with distance from the finite end of the cylinder. [Copyright &y& Elsevier]

Abstract: We study the existence of radial ground state solutions for the problem , . It is known that this problem has infinitely many ground states when , while no solutions exist if . A question raised by Ni and Serrin in [W.-M. Ni, J. Serrin, Existence and non-existence theorems for ground states for quasilinear partial differential equations, Atti Convegni Lincei 77 (1985) 231–257] is whether or not ground state solutions exist for . In this paper we prove the existence of a large, finite number of ground states with fast decay as provided that q lies below but close enough to the critical exponent . These solutions develop a bubble-tower profile as q approaches the critical exponent. [Copyright &y& Elsevier]

Abstract: We study a large time behavior of a solution to the initial boundary value problem for an isentropic and compressible viscous fluid in a one-dimensional half space. The unique existence and the asymptotic stability of a stationary solution are proved by S. Kawashima, S. Nishibata and P. Zhu for an outflow problem where the fluid blows out through the boundary. The main concern of the present paper is to investigate a convergence rate of a solution toward the stationary solution. For the supersonic flow at spatial infinity, we obtain an algebraic or an exponential decay rate. Precisely, if an initial perturbation decays with the algebraic or the exponential rate in the spatial asymptotic point, the solution converges to the corresponding stationary solution with the same rate in time as time tends to infinity. An algebraic convergence rate is also obtained for the transonic flow. These results are proved by the weighted energy method. [Copyright &y& Elsevier]

Abstract: In this paper we consider the one-dimensional p-Laplacian boundary value problem on time scales where is p-Laplacian operator, i.e., , . Some new results are obtained for the existence of at least single, twin or triple positive solutions of the above problem by using Krasnosel''skii''s fixed point theorem, new fixed point theorem due to Avery and Henderson and Leggett–Williams fixed point theorem. This is probably the first time the existence of positive solutions of one-dimensional p-Laplacian boundary value problems on time scales has been studied. [Copyright &y& Elsevier]

Abstract: In this paper, we give a global well-posedness result for the two-dimensional Boussinesq system with partial viscosity, when the initial data and . [Copyright &y& Elsevier]

Abstract: In this paper, some sufficient conditions for the existence of solutions to the boundary value problems of a class of second order difference equation are obtained by using the critical point theory. [Copyright &y& Elsevier]

Abstract: In this paper, we discuss an initial-boundary value problem and a Cauchy problem for the stochastic porous medium equation. Our basic estimates are based on the known results due to (Math. Sbornik 67 (1965) 609–642, Arch. Rational Mech. Anal. 25 (1967) 64–80). By the procedure developed in (Trans. Amer. Math. Soc. 354 (2001) 1117–1135), we obtain solutions over the given probability space rather than martingale solutions. We will also establish the existence of invariant measures when the space domain is bounded. [Copyright &y& Elsevier]

Abstract: This paper is devoted to study the following third-order multi-point singularly perturbed boundary value problemwhere , , , , and . The existence, uniqueness and asymptotic estimates of solutions of the boundary value problem are give by using priori estimates, differential inequalities technique and Leray–Schauder degree theory. [Copyright &y& Elsevier]

Abstract: In this paper we consider boundary-value problems in domains with perforated boundaries. We use the classification of homogenized (limit) problems depending on the ratio of small parameters, which characterize the diameter of the holes and the distance between them. We study the analogue of the Helmholtz resonator for domains with a perforated boundary. [Copyright &y& Elsevier]

In this paper, we consider the local existence of solutions to Euler equations with linear damping under the assumption of physical vacuum boundary condition. By using the transformation introduced in Lin and Yang (Methods Appl. Anal. 7 (3) (2000) 495) to capture the singularity of the boundary, we prove a local existence theorem on a perturbation of a planar wave solution by using Littlewood–Paley theory and justifies the transformation introduced in Liu and Yang (2000) in a rigorous setting. [Copyright &y& Elsevier]

Abstract: In this paper, we study the vanishing viscosity limit of initial boundary value problems for one-dimensional mixed hyperbolic–parabolic systems when the boundary is characteristic for both the viscous and the inviscid systems: in particular, we assume that an eigenvalue of the inviscid system vanishes uniformly. We prove the stability of boundary layers expansions in small time (i.e before shocks for the inviscid system) as long as the amplitude of the boundary layers remains sufficiently small. In particular, by using Lagrangian coordinates, we apply our result to physical systems like gasdynamics and magnetohydrodynamics with homogeneous Dirichlet condition for the velocity at the boundary. [Copyright &y& Elsevier]