This work is devoted to the study of traveling semifronts for an SIS epidemic model in a patchy environment. In the SIS epidemic model, infective individuals become susceptible again after recovery, thereby forming a cyclic structure from the infected class back to the susceptible class. The existence of traveling semifronts of the system is established via Schauder's fixed point theorem with the aid of a pair of upper and lower solutions distinct from those in SI-type epidemic models. We further verify the persistence of these traveling semifronts by virtue of the boundedness of the second component, a core prerequisite for establishing the existence of critical-speed traveling semifronts. To address the obstacles posed by the cyclic structure and linear growth of the second component, we employ Harnack-type properties for its corresponding traveling wave equation, together with a series of refined analytical arguments, to establish the boundedness of the traveling semifronts. Moreover, traveling waves connecting the disease-free equilibrium to the endemic equilibrium of the system with a specific demographic structure for the susceptible group and a bilinear incidence function are achieved by constructing a suitable Lyapunov function. Finally, the convergence of persistent positive traveling semifronts at +infinity for the original system is illustrated by numerical simulations.
This work explores the dynamics of nonlo cal reaction-diffusion equations under nonlinear growth conditions in thin domains. The nonlo cal effect appears in a viscosity term in front of the Laplacian operator. Existence of strong solutions and global attractors is ensured. By identifying a limit equation on the lower-dimensional domain, we compare the associated family of attractors, revealing upper-semicontinuity.
We study the existence and multiplicity of nontrivial homo clinic solutions for a class of periodic discrete nonlinear Schro & uml;dinger (DNLS) equations when the nonlinearities can be local superlinear or local asymptotically linear at infinity by means of the critical point theory. Principal challenges of the resulting problem entail the restriction of local growth conditions and the boundedness of Cerami sequences. We incorporate weak*-compactness for bounded sequences, unique continuation of discrete Schro & uml;dinger equations, and Lusternik-Schnirelman theory under symmetries to solve the problem. This allows us for the first time to derive a ground state and infinitely many homoclinic solutions with such assumptions, improving the relevant works in the literature. Our result implies the existence of infinitely many exponentially localized states of periodic discrete Schro & uml;dinger equations with nonlinear impurity. Our work also answers a problem of A. Pankov from the 2010's on the existence of nontrivial homo clinic solutions for periodic DNLS equations.
In this paper, we generalize the Ollivier-Ricci curvature lower bound in the W1-Wasserstein metric sense from discrete time to continuous time jumping Markov processes. This becomes more flexible when the generator equals J-I than Ollivier's curvature of a transition probability kernel J. We show that this lower bound of the Ollivier-Ricci curvature can be characterized by some optimal coupling generator, and we provide the construction of this by introducing the "false" jumps (Villemonais has constructed such a coupling without proving its optimality). Some previous results of Ollivier for discrete-time Markov chains are generalized to the actual continuous-time case, and the exponential convergence in entropy is derived by using recent progress made by Caputo-Mu & uml;nch-Salez on the Peres-Tetali conjecture. Furthermore, we propose a new comparison condition with some birth-death process on N to obtain some explicit exponential convergence rate by modifying the metric, even if the Ollivier-Ricci curvature with respect to the graph metric is not positive. As an application, we generalize the result of Eberle, Luo, and Wang for the W1-exponential convergence of dissipative-at-infinity diffusions to the jumps processes on graphs. Finally, we generalize Hairer-Mattingly's theorem from Markov chains to continuous time Markov processes.
This paper investigates the spectral stability of boundary layer profiles for the two-dimensional inhomogeneous Navier-Stokes equations in the high-Reynolds-number limit. We identify a "generalized concavity" defined as (rho U-s(s)')' < 0, which characterizes the interplay between density stratification and velocity shear. Under this structural assumption, we establish the existence of Tollmien-Schlichting instabilities and prove spectral stability of background shear flow within the optimal Gevrey 3/2 class. These results extend the classical stability theory of homogeneous boundary layers to the stratified setting and capture influence of density variations on fluid stability.
For the parabolic k-Hessian equation -u(t)sigma(k)(D(2)u) = 1, Cao and Jiang [4] constructed a new class of entire solutions of variable separated form. We denote those solutions by H-k and use H-k to perturb the right-hand side of the equation, consider -u(t)sigma(k)(D(2)u) = f with f = 1 + O(H-k(-beta/2)) for 3 > 2 and prove the following results. (i) The existence of parabolically k-convex viscosity entire solutions with prescribed asymptotic behavior at infinity like u = H-k+ O(H-k(1- gamma)) where gamma = min{(k + 1)/C-n,C-k, 3/2}. (ii) The existence of parabolically k-convex viscosity solutions on exterior domains Rn+1\(D) over bar for bowl-shaped D, with Dirichlet boundary data phi and asymptotic behavior like u = H-k+ c + O(H-k(1-gamma)), c >= c* for some c*. The asymptotic behavior at infinity of u is quite different from the case that-t + P(x) is the main term as studied by Zhou and Bao [12]. Our results both rely on Perron method, and generalize the work of Bao et al. [3] from parabolic Monge-Ampere equations to parabolic k-Hessian equations.
Our aim in this review paper is to discuss the well-posedness of the Cahn-Hilliard equation, as well as of several of its important variants, with logarithmic nonlinear terms. More precisely, we also address the Cahn-Hilliard equation with a source term, Cahn-Hilliard models based on a microforce balance, and a new first-order hyperbolic Cahn-Hilliard model. Furthermore, we discuss the so-called strict separation property of the solutions from the singular values of the nonlinear term. This property is only valid in one and two space dimensions in general and allows to study additional properties of the solutions in a simpler way, since we then essentially have a regular and bounded nonlinear term. These results are new for the first-order hyperbolic Cahn-Hilliard model and improve previous ones for the Cahn-Hilliard equation with a source term, allowing us to consider a larger class of source terms (some specific examples of source tems are also given).
This study investigates a chemotaxis system with acceleration effects and logistic growth, governed by the following coupled PDEs on a smooth bounded domain ohm subset of Rn(n >= 1): { u(t) = triangle u - del & centerdot; (uw) + gamma(u - u (alpha)), x is an element of ohm, t > 0, v(t) = triangle v - vu, x is an element of ohm, t > 0, w(t )= triangle w - w + chi del v, x is an element of ohm, t > 0 with homogeneous Neumann conditions for u, v and Dirichlet conditions for w. Parameters satisfy chi > 0, gamma >= 0 and alpha > 1. By employing L-p-estimates and a bootstrap iteration scheme, we analyze the system's nonlinear dynamics and establish two key results: (i) Global boundedness: Solutions exist globally and remain bounded if either n <= 3 with gamma >= 0 and alpha > 1, or n >= 4 with gamma > 0 and alpha > n+6-root n(2)+4n+20 /2 ; (ii) Asymptotic behavior: For small chi, solutions converge exponentially to the steady state (m, 0, 0), where m = 1/ |ohm| integral(ohm)u(0)(gamma = 0) or m = 1(gamma > 0) Our analysis reveals how acceleration and logistic terms interplay to regulate solution bounds and long-term dynamics, generalizing classical chemotaxis models. The results are rigorously validated through dimension-specific thresholds and sensitivity conditions.
. We investigate the resonance problem for a planar Hamiltonian system Jz ' = del H(t, z) under Landesman-Lazer conditions imposed at both zero and infinity. We compare the behavior of its solutions with those of two appropriate positively homogeneous systems at zero and infinity, and establish a multiplicity result when the gap between the rotation numbers of the referred positively homogeneous systems at zero and infinity vanishes. Our approach is based on the Poincare-Birkhoff theorem and the expression of the rotational characterizations of Landesman-Lazer conditions.
In this study, we investigate the inverse scattering transform for the vector (n-component) derivative nonlinear Schro & uml;dinger equation, addressing technical challenges arising from N pairs of higher-order poles in the transmission coefficient. To resolve the complexities inherent in multi-component systems, we introduce a generalized cross product for vectors in Cn+1, which clarifies the intricate correlations between the relevant derivatives of modified eigenfunctions at the discrete spectrum. Further, we formulate an (n + 1) & times; (n + 1) matrix-valued Riemann-Hilbert problem and propose two integral factors to ensure it meets the normalization condition. Subsequently, we prove the existence and uniqueness of solutions for an equivalent problem free of singularities. Finally, we explicitly reconstruct the multiple-pole soliton solutions under reflectionless conditions.
. This paper deals with the perturbations of piecewise smooth integrable non-Hamiltonian systems separated by a nonregular line. We develop the algorithms of the Melnikov functions of arbitrary order to analyze limit cycles that can bifurcate from the periodic annulus. While previous studies have explored the Melnikov functions of arbitrary order for perturbations of piecewise Hamiltonian systems separated by a straight line, our work extends to piecewise smooth integrable non-Hamiltonian systems with a nonregular line. As applications, lower bounds for the number of limit cycles are provided for two concrete systems.
