Abstract: Publication date: June 2020Source: Nonlinear Analysis: Real World Applications, Volume 53Author(s): Andrea Giacobbe, Giuseppe Mulone, Wendi Wang A mathematical model is proposed to simulate the eating disorders of bulimia or anorexia. Earlier models are extended to incorporate the body mass index, which plays a key role in the eating attitude of self thinners. The global existence and ultimate boundedness of solutions of the nonlocal model are proved by using estimates of solutions. The basic reproduction number of eating disorder contagion is shown to be the invasion threshold. The testable linear and nonlinear stability conditions are established by Lyapunov functions. Further numerical simulations are given to reveal how self-forces and peer pressures to be thinner affect the emergence and distributions of eating disorders.

Abstract: Publication date: June 2020Source: Nonlinear Analysis: Real World Applications, Volume 53Author(s): Stephen Baigent, Belgin Seymenoğlu We study the two-locus-two-allele (TLTA) Selection–Recombination model from population genetics and establish explicit bounds on the TLTA model parameters for an invariant manifold to exist. Our method for proving existence of the invariant manifold relies on two key ingredients: (i) monotone systems theory (backwards in time) and (ii) a phase space volume that decreases under the model dynamics. To demonstrate our results we consider the effect of a modifier gene β on a primary locus α and derive easily testable conditions for the existence of the invariant manifold.

Abstract: Publication date: June 2020Source: Nonlinear Analysis: Real World Applications, Volume 53Author(s): Y. Paulina Martínez, Claudia Valls We study a Chemostat system of the form ẋ=−qx+R̃K+yxy,ẏ=(c̃−y)q−R̃ã(K+y)xy,where q>0, R̃>0, K>0, c̃>0 and ã≠0. This system appears in competition modelling in biology. We describe its global dynamics on the Poincaré disc and study its Liouvillian integrability. For the first topic we use the well-known Poincaré compactification theory and for the second one we make use of the Puiseux series to derive the structure of all the irreducible invariant algebraic curves.

Abstract: Publication date: April 2020Source: Nonlinear Analysis: Real World Applications, Volume 52Author(s): Weihua Wang Consider the stochastic Navier–Stokes–Coriolis equation in R3 driven by an additive white noise, we establish the unique existence and spatial analyticity of global mild solution even when initial data is essentially arbitrarily large and even when stochastic external force is also large provided that the speed of the rotation is fast enough. The proof is based on the Picard contraction principle and a priori estimates to the stochastic parabolic equation.

Abstract: Publication date: April 2020Source: Nonlinear Analysis: Real World Applications, Volume 52Author(s): Jihua Yang, Erli Zhang By analyzing the corresponding Picard–Fuchs equations, we obtain an upper bound of the number of limit cycles for a class of piecewise smooth Hamiltonian systems when they are perturbed inside discontinuous polynomials of degree n. Finally, we present an example to illustrate an application of the theoretical results.

Abstract: Publication date: April 2020Source: Nonlinear Analysis: Real World Applications, Volume 52Author(s): Xi Wei, Guangsheng Wei, Feng-Bin Wang, Hua Nie This study proposes and analyzes a reaction–diffusion system describing the competition of two species for a single limiting nutrient that is stored internally in an unstirred chemostat, in which each species also produces a toxin that increases the mortality of its competitors. The possibility of coexistence and bistability for the model system is studied by the theory of uniform persistence and topological degree theory in cones, respectively. More precisely, the sharp a priori estimates for nonnegative solutions of the system are first established, which assure that all of nonnegative solutions belong to a special cone. Then it turns out that coexistence and bistability can be determined by the sign of the principal eigenvalues associated with specific nonlinear eigenvalue problems in the special positive cones. The local stability of two semi-trivial steady states cannot be studied via the technique of linearization since a singularity arises from the linearization around those steady states. Instead, we introduce a 1-homogeneous operator to rigorously investigate their local stability.

Abstract: Publication date: April 2020Source: Nonlinear Analysis: Real World Applications, Volume 52Author(s): Kaiqiang Li, Weike Wang In this paper, we are interested in the well-posedness of solutions for anisotropic scalar conservation law with large perturbation around the planar shock wave in two dimensional space. It is shown that the Cauchy problem admits a global classical solution by using the classical energy estimates and contraction property, while the initial perturbation is bounded in given space, where we used some new inequalities for anisotropic terms which was imposed in Lemma 2.7. Moreover, we obtain the pointwise estimates of the solutions for fractional order dissipation in y-direction by the formula of Green’s function.

Abstract: Publication date: April 2020Source: Nonlinear Analysis: Real World Applications, Volume 52Author(s): Shouqiong Sheng, Zhiqiang Shao In this paper, we study the limiting behavior of Riemann solutions to the Euler equations of one-dimensional compressible fluid flow as γ tends to one. We show that the limit solution forms the delta wave to the pressureless Euler system of one-dimensional compressible fluid flow in the distribution sense. Some numerical results exhibiting the phenomena of concentration are also presented.

Abstract: Publication date: April 2020Source: Nonlinear Analysis: Real World Applications, Volume 52Author(s): M.L. Santos, M.M. Freitas, A.J.A. Ramos In this paper we study the long-time behavior of binary mixture problem of solids, focusing on the interplay between nonlinear damping and source terms. By employing nonlinear semigroups and the theory of monotone operators, we obtain several results on the existence of local and global weak solutions, and uniqueness of weak solutions. Moreover, we prove that every weak solution to our system blows up in finite time, provided the initial energy is negative and the sources are more dominant than the damping in the system. Additional results are obtained via careful analysis involving the Nehari Manifold. Specifically, we prove the existence of a unique global weak solution with initial data coming from the “good” part of the potential well. For such a global solution, we prove that the total energy of the system decays exponentially or algebraically, depending on the behavior of the dissipation in the system near the origin.

Abstract: Publication date: April 2020Source: Nonlinear Analysis: Real World Applications, Volume 52Author(s): Ngul Suan Lian, Kai Yan This paper is devoted to the study of persistence properties and asymptotic behavior for a two-component b-family system with high order nonlinearity. We prove that both the density and momentum components of the corresponding solutions with initial compact support will retain the property of being compactly supported throughout its evolution. Moreover, we investigate the asymptotic behaviors of the solutions at infinity within its lifespan when the initial data decay exponentially and algebraically, respectively.

Abstract: Publication date: April 2020Source: Nonlinear Analysis: Real World Applications, Volume 52Author(s): Jianjun Chen, Gan Yin, Shouke You This paper is concerned with the process of a gas expanding into vacuum by turning around a sharp corner for 2-D pseudo-steady compressible magnetohydrodynamics system. This problem actually can be interpreted as interaction of a centered wave and a planar rarefaction wave. As the gas touches the corner and starts expanding into the vacuum around the sharp corner, a centered wave and a planar rarefaction wave appear. In the estimates of solution, we utilize the characteristic analysis and deduce proper characteristic decompositions. Combining the C0 estimate, gradient estimates, hyperbolicity, and the interaction of centered wave and planar rarefaction wave, we constructively obtain the existence of global classical solution to the present problem.

Abstract: Publication date: April 2020Source: Nonlinear Analysis: Real World Applications, Volume 52Author(s): Cristina Lois-Prados, Radu Precup The paper deals with a non-autonomous Lotka–Volterra type system, which in particular may include logistic growth of the prey population and hunting cooperation between predators. We focus on the existence of positive periodic solutions by using an operator approach based on the Krasnosel’skii homotopy expansion theorem. We give sufficient conditions in order that the localized periodic solution does not reduce to a steady state. Particularly, two typical expressions for the functional response of predators are discussed.

Abstract: Publication date: April 2020Source: Nonlinear Analysis: Real World Applications, Volume 52Author(s): Mario Fuest The Neumann initial–boundary problem for the chemotaxis system (⋆)ut=Δu−∇⋅(u∇v)+κ( x )u−μ( x )up,0=Δv−m(t) Ω +u,m(t)≔∫Ωu(⋅,t)is studied in a ball Ω=BR(0)⊂R2, R>0 for p≥1 and sufficiently smooth functions κ,μ:[0,R]→[0,∞).We prove that whenever μ′,−κ′≥0 as well as μ(s)≤μ1s2p−2 for all s∈[0,R] and some μ1>0 then for all m0>8π there exists u0∈C0(Ω¯) with ∫Ωu0=m0...

Abstract: Publication date: April 2020Source: Nonlinear Analysis: Real World Applications, Volume 52Author(s): Dugyu Kim Consider the stationary motion of an incompressible Navier–Stokes fluid around a rotating body R3∖Ω which is also moving in the direction of the axis of rotation with nonzero constant velocity −ke1. We assume that the angular velocity ω= ω e1 is also constant and the external force is given by f=divF. Then the motion is described by a variant of the stationary Navier–Stokes equations with the velocity ke1 at infinity. Our main result is the existence of at least one solution u satisfying u−ke1∈L3(Ω) for arbitrarily large F∈L3∕2(Ω). The uniqueness is also proved by assuming that ω + k +‖F‖L3∕2(Ω) is sufficiently small in comparison with the viscosity ν. Moreover, we establish several regularity results to obtain an existence theorem for weak solutions u satisfying ∇u∈L3∕2(Ω) and u−ke1∈L3(Ω).