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Abstract: In this paper, we study the dynamics of predator–prey interaction systems between two species with ratio-dependent functional responses. First we provide sufficient and necessary conditions for positive steady-state solutions, and then we investigate the relationships between positive equilibria and positive solutions of the system over a large domain. Furthermore, we deal with the uniqueness and the stability of positive steady-states solutions with some assumptions. In addition, we discuss the extinction and the persistence results of time-dependent positive solutions to the system. [Copyright &y& Elsevier]

Large number of papers have been devoted to the study of tumor models. Wellpostedness as well as properties of solutions are systematically studied [1–3,5–21,25–40]. The properties include asymptotic behavior, stability, bifurcation analysis, etc. In this paper we study a tumor model with a time-delay, and establish a bifurcation result for all even mode n ≥ 2. The time delay represents the time taken for cells to undergo replication (approximately 24 hours). In contrast to some results in the literature where bifurcation is established for sufficiently large bifurcation parameter, our result includes the smallest bifurcation point, which is crucial as this is the point of stability changes under certain conditions as we have shown in [40]. The inclusion of a time delay, although biologically very reasonable, introduces two significant mathematical challenges: (a) the explicit solution utilized to verify the bifurcation theorem is no longer available; (b) the system becomes non-local because of the time-delay, which in term produces technical difficulties for the PDE estimates. To our knowledge, this is the first paper on the study of bifurcation for tumor models with a time delay. [ABSTRACT FROM AUTHOR]

In this paper we discuss existence of pointwise solutions for dynamical models of viscoplasticity. Among other things, this work answers the question about necessity of safe-load conditions in case of viscoplasticity, which arise in the paper of K. Chełmiński (2001) [11]. We proved that solutions can be obtained without assuming any kind of safe-load conditions. Moreover, in the manuscript we consider much more general model than in the above mentioned paper. Namely, we consider the model with mixed boundary conditions and we allow a possible disturbance of the inelastic constitutive function by a globally Lipschitz function. Presented approach shows that via the same methods one can prove existence of pointwise solutions for: coercive models, self-controlling models, models with polynomial growth (not necessary of single valued) and monotone-gradient type models of viscoplasticity. [ABSTRACT FROM AUTHOR]

The aim of this paper is characterizing the development of singularities by the positive solutions of the quasilinear indefinite Neumann problem − (u ′ / 1 + (u ′) 2 ) ′ = λ a (x) f (u) in (0 , 1) , u ′ (0) = 0 , u ′ (1) = 0 , where λ ∈ R is a parameter, a ∈ L ∞ (0 , 1) changes sign once in (0 , 1) at the point z ∈ (0 , 1) , and f ∈ C (R) ∩ C 1 [ 0 , + ∞) is positive and increasing in (0 , + ∞) with a potential, ∫ 0 s f (t) d t , superlinear at +∞. In this paper, by providing a precise description of the asymptotic profile of the derivatives of the solutions of the problem as λ → 0 + , we can characterize the existence of singular bounded variation solutions of the problem in terms of the integrability of this limiting profile, which is in turn equivalent to the condition (∫ x z a (t) d t) − 1 2 ∈ L 1 (0 , z) and (∫ x z a (t) d t) − 1 2 ∈ L 1 (z , 1). No previous result of this nature is known in the context of the theory of superlinear indefinite problems. [ABSTRACT FROM AUTHOR]

Delays are ubiquitous, pervasive, and entrenched in everyday life, thus taking it into consideration is necessary. Dupire recently developed a functional Itô formula, which has changed the landscape of the study of stochastic functional differential equations and encouraged a reconsideration of many problems and applications. Based on the new development, this work examines functional diffusions with two-time scales in which the slow-varying process includes path-dependent functionals and the fast-varying process is a rapidly-changing diffusion. The gene expression of biochemical reactions occurring in living cells in the introduction of this paper is such a motivating example. This paper establishes mixed functional Itô formulas and the corresponding martingale representation. Then it develops an averaging principle using weak convergence methods. By treating the fast-varying process as a random "noise", under appropriate conditions, it is shown that the slow-varying process converges weakly to a stochastic functional differential equation whose coefficients are averages of that of the original slow-varying process with respect to the invariant measure of the fast-varying process. [ABSTRACT FROM AUTHOR]

In this paper, we use the Gromov-Hausdorff distances between two global attractors (which belong to disjoint phase spaces) and two dynamical systems to consider the continuous dependence of the global attractors and the stability of the dynamical systems on global attractors induced by the reaction diffusion equation under perturbations of the domain. The novelty of the paper is to compare any two systems in different phase spaces without the process of "pull-backing" the perturbed systems to the original domain. [ABSTRACT FROM AUTHOR]

The paper investigates the well-posedness and longtime dynamics of the Kirchhoff wave model with structural nonlinear damping: u t t − ϕ (‖ ∇ u ‖ 2) Δ u + σ (‖ ∇ u ‖ 2) (− Δ) θ u t + f (u) = g (x) , with θ ∈ [ 1 / 2 , 1). We find a new critical exponent p ⁎ ≡ N + 2 N − 2 (> p θ ≡ N + 2 θ N − 2 , N ≥ 3) and show that when the growth exponent p of the nonlinearity f (u) is up to the range: 1 ≤ p < p ⁎ : (i) the IBVP of the equation is well-posed and its solution is of additionally global regularity when t > 0 ; (ii) for each θ ∈ [ 1 / 2 , 1) , the related solution semigroup has in natural energy space H an optimal global attractor A θ whose compactness and attractiveness are in the regularized space H 1 + θ where A θ lies, and an optimal exponential attractor E θ ⁎ whose compactness, boundedness of the fractional dimension and the exponential attractiveness are in H 1 + θ where E θ ⁎ lies, respectively; (iii) the family of global attractors { A θ } θ ∈ [ 1 / 2 , 1) is upper semi-continuous at each point θ 0 ∈ [ 1 / 2 , 1). The paper breaks though the longstanding existed growth restriction: 1 ≤ p ≤ p θ for p θ had been considered a uniqueness index, deepens and extends the results in literature [6,25,27]. [ABSTRACT FROM AUTHOR]

In this paper, we investigate multiple existence of homoclinic solutions for a periodically perturbed N -dimensional autonomous differential equation with a degenerate homoclinic solution of degeneracy degree d. Known results were obtained with a functional perturbation, which is regarded as an infinite-dimensional parameter. In this paper we consider a single parameter perturbation, a special form of former's functional perturbation, and prove that the single parameter is enough to unfold all possibilities of linearly independent homoclinic solutions bifurcated from the unperturbed degenerate homoclinic one, which actually improves the known results. Furthermore, we prove that those homoclinic solutions are all transversal, showing co-existence of multiple chaotic motions. [ABSTRACT FROM AUTHOR]

This paper is concerned with the large time behaviors of the entropy solutions to one-dimensional scalar convex conservation laws, of which the initial data are assumed to approach two arbitrary L ∞ periodic functions as x → − ∞ and x → + ∞ , respectively. We show that the solutions approach the Riemann solutions at algebraic rates as time increases. Moreover, a new discovery in this paper is that the difference between the two periodic perturbations at two infinities may generate a constant shift on the background shock wave, which is different from the result in [11] , where the two periodic perturbations are the same. [ABSTRACT FROM AUTHOR]

We investigate spectral theory for a one-body Stark Hamiltonian under minimum regularity and decay conditions on the potential (actually allowing sub-linear growth at infinity). Our results include Rellich's theorem, the limiting absorption principle, radiation condition bounds and Sommerfeld's uniqueness, and most of these are stated and proved in sharp form employing Besov-type spaces. For the proofs we adopt a commutator scheme by Ito–Skibsted [13]. A feature of the paper is a special choice of an escape function related to parabolic coordinates, which conforms well with classical mechanics for the Stark Hamiltonian. The whole setting of the paper, such as the conjugate operator and the Besov-type spaces, is generated by this single escape function. We apply our results in the sequel paper [5]. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to investigate mathematically gas discharge. Townsend discovered α - and γ -mechanisms which are essential for ionization of gas, and then derived a threshold of voltage at which gas discharge can happen. In this derivation, he used some simplification such as discretization of time. Therefore, it is an interesting problem to analyze the threshold by using the Degond–Lucquin-Desreux–Morrow model and also to compare the results of analysis with Townsend's theory. Note that gas discharge never happens in Townsend's theory if γ -mechanism is not taken into account. In this paper, we study an initial–boundary value problem to the model with α -mechanism but no γ -mechanism. This problem has a trivial stationary solution of which the electron and ion densities are zero. It is shown that there exists a threshold of voltage at which the trivial solution becomes unstable from stable. Then we conclude that gas discharge can happen for a voltage greater than this threshold even if γ -mechanism is not taken into account. It is also of interest to know the asymptotic behavior of solutions to this initial–boundary value problem for the case that the trivial solution is unstable. To this end, we establish bifurcation of non-trivial stationary solutions by applying Crandall and Rabinowitz's Theorem, and show the linear stability and instability of those non-trivial solutions. [ABSTRACT FROM AUTHOR]

This paper is concerned with the validity of the Prandtl boundary layer theory in the inviscid limit of the steady incompressible Navier-Stokes equations, which is an extension of the pioneer paper [13] (Y. Guo et al., 2017, Ann. PDE) from a domain of [ 0 , L ] × R + to [ 0 , L ] × [ 0 , 2 ]. Under the symmetry assumption, we establish the validity of the Prandtl boundary layer expansions and the error estimates. The convergence rate as ε → 0 is also given. [ABSTRACT FROM AUTHOR]

This paper studies a new gradient regularity in Lorentz spaces for solutions to a class of quasilinear divergence form elliptic equations with nonhomogeneous Dirichlet boundary conditions: { div (A (x , ∇ u)) = div (| F | p − 2 F) in Ω , u = σ on ∂ Ω , where Ω ⊂ R n (n ≥ 2), the nonlinearity A is a monotone Carathéodory vector valued function defined on W 0 1 , p (Ω) for p > 1 and the p -capacity uniform thickness condition is imposed on the complement of our bounded domain Ω. Moreover, for given data F ∈ L p (Ω ; R n) , the problem is set up with general Dirichlet boundary data σ ∈ W 1 , p (Ω). In this paper, the optimal good- λ type bounds technique is applied to prove some results of fractional maximal estimates for gradient of solutions. And the main ingredients are the action of the cut-off fractional maximal functions and some local interior and boundary comparison estimates developed in previous works [45,52,53] and references therein. [ABSTRACT FROM AUTHOR]

This paper establishes the global regularity of the two-dimensional (2D) Boussinesq equations and micropolar equations with partial dissipation. Our first result is the global regularity of the 2D Boussinesq equations with fractional vertical dissipation in the horizontal velocity, horizontal dissipation in the vertical velocity and zero thermal diffusion, which is shown by taking advantage of the nice structure of the 2D Boussinesq equations and several refined commutator estimates. The second goal of this paper is to consider a system of the 2D incompressible micropolar equations with vertical dissipation in the horizontal velocity equation, horizontal dissipation in the vertical velocity equation and the fractional Λ α dissipation in the micro-rotation velocity. In order to overcome the difficulty caused by the lack of full Laplacian diffusion in the velocity equations, we fully exploit the nice structure of the corresponding equations to show that this equations with arbitrarily small α > 0 always possesses a unique global classical solution. [ABSTRACT FROM AUTHOR]

In this paper, we consider a model for the spin-magnetization system that takes into account the diffusion process of the spin accumulation. This model consists of the Landau-Lifshitz equation describing the precession of the magnetization, coupled with a quasi-linear parabolic equation describing the diffusion of the spin accumulation. This paper establishes the global existence and uniqueness of weak solutions for large initial data in R 2. Moreover, partial regularity is shown. In particular, the solution is regular on R 2 × (0 , ∞) with the exception of at most finite singular points. [ABSTRACT FROM AUTHOR]

This paper establishes nonlinear asymptotic stability of homogeneous reference states in dissipative relativistic fluid dynamics. The result is a counterpart for general non-barotropic fluids of one obtained by the author in a previous paper on barotropic fluids. Differently from that of this earlier finding, the proof here crucially relies on analyzing the corresponding linearized problem in Fourier space, with different scalings for small and large wave numbers. [ABSTRACT FROM AUTHOR]

This paper concerns the exact bound on the number of zeros of Abelian integrals associated with two hyper-elliptic Hamiltonian systems of degree 4. The upper and lower bounds for the two systems have been obtained in several previous works, however the sharp bounds are still unknown. In this paper, we provide a proof to show that the exact bound is 3 for both systems. The basic idea of our method is to bound the parameter space in R 3 to obtain two parameter sets, which might yield maximal 4 zeros of Abelian integrals corresponding to the two sets. Further, we show that the existence of 4 zeros on the two sets is not possible, and thus the sharp bound is 3. [ABSTRACT FROM AUTHOR]

In this paper, we consider the initial-boundary value problem to the compressible Navier-Stokes equations for ideal gases without heat conduction in the half space or outside a fixed ball in R N , with N ≥ 1. We prove that any classical solutions (ρ , u , θ) , in the class C 1 ([ 0 , T ] ; H m (Ω)) , m > [ N 2 ] + 2 , with bounded from below initial entropy and compactly supported initial density, which allows to touch the physical boundary, must blow-up in finite time, as long as the initial mass is positive. This paper extends the classical result by Xin (1998) [19] , in which the Cauchy problem is considered, to the case that with physical boundary. [ABSTRACT FROM AUTHOR]

To incorporate spatial memory and nonlocal effect of animal movements, we propose and investigate the spatiotemporal dynamics of the single population model with memory-based diffusion and nonlocal reaction. We first study the stability of a positive equilibrium and the steady state bifurcation induced by diffusion and nonlocality. We then investigate the impact of the averaged memory period on stability and bifurcation, and show that the combination of the averaged memory period and the diffusion can lead to the occurrence of Turing-Hopf and double Hopf bifurcations. The paper originally derives the normal form theory for Turing-Hopf bifurcation in the general reaction-diffusion equation with memory-based diffusion and nonlocal reaction. This novel algorithm can be widely used to classify the spatiotemporal dynamics near the Turing-Hopf bifurcation point. Finally, we apply the obtained results to a model proposed by Britton and numerically illustrate the spatiotemporal patterns induced by Hopf, Turing-Hopf and double Hopf bifurcations. Stable spatially homogeneous/nonhomogeneous periodic solutions, homogeneous/nonhomogeneous steady states and the transition from one of these solutions to another are provided in this paper. We additionally acquire the coexistence of two stable spatially nonhomogeneous steady states or two spatially nonhomogeneous periodic solutions near the Turing-Hopf bifurcation point. • Propose the single population model with memory-based diffusion and nonlocal reaction. • Find the codimension-two Turing-Hopf and double Hopf bifurcation in the scalar reaction-diffusion equation. • Develop the normal form of Turing-Hopf bifurcation in the equation with memory-based diffusion and nonlocal term. • Find the coexistence of two stable spatially inhomogeneous steady states or two spatially inhomogeneous periodic solutions. • Find the transition from one of unstable solutions to another stable one. [ABSTRACT FROM AUTHOR]

This paper is devoted to the mathematical theory of two-dimensional injection of incompressible, irrotational, and inviscid fluids issuing from two infinitely long nozzles into a free stream. In general, there is a free interface with constant jump of the Bernoulli constant on it, which is different and more difficult than the previous related works. Physically, it is called the collision fluid. The main result in this paper is that for given two co-axis symmetric infinitely long nozzles, imposing the incoming mass fluxes in the two nozzles, there exists a unique piecewise smooth collision fluid, such that the free interface of the collision fluid is a C 1 curve, and the pressure is continuous across the interface. As byproducts, the asymptotic behaviors, the positivity of the vertical velocity, monotone relationship between the location of the interface and the incoming mass fluxes are also established. [ABSTRACT FROM AUTHOR]

The current paper is devoted to the Cauchy problem for the white noise driven Ostrovsky equation with positive dispersion. We obtain the local well-posedness for the initial data u 0 (⋅ , ω) ∈ H s (R) (a.e. ω ∈ Ω) which is F 0 -measurable with s > − 3 4 and Φ ∈ L 2 0 , s ∩ L 2 0 (L 2 (R) , H ˙ s , − 1 2 + ϵ (R)) , and the globally well-posedness for the initial data u 0 (⋅ , ω) ∈ L 2 (R) (a.e. ω ∈ Ω) which is F 0 -measurable and Φ ∈ L 2 0 , 0 ∩ L 2 0 (L 2 (R) , H ˙ 0 , − 1 2 + ϵ (R)). The key ingredients that we used in this paper are some bilinear estimates in X s , b -type spaces given in subsection 2.1 , Itô formula, the BDG inequality and stopping time techniques as well as frequency truncated technique. Our method can be applied to the case γ = 0 , which is just the KdV equation, thus, our result improves the local well-posedness result of A. de Bouard, A. Debussche and Y. Tsutsumi (1999) [3]. [ABSTRACT FROM AUTHOR]

This paper deals with the quasilinear degenerate chemotaxis system with flux limitation, { u t = ∇ ⋅ ( u p ∇ u u 2 + | ∇ u | 2 ) − χ ∇ ⋅ ( u q ∇ v 1 + | ∇ v | 2 ) , 0 = Δ v − μ + u , under no-flux boundary conditions in balls Ω ⊂ R n , and the initial condition u | t = 0 = u 0 for a radially symmetric and positive initial data u 0 ∈ C 3 (Ω ‾) , where χ > 0 and μ : = 1 | Ω | ∫ Ω u 0. Bellomo–Winkler [3] proved local existence of unique classical solutions and extensibility criterion ruling out gradient blow-up as well as global existence and boundedness of solutions when p = q = 1 under some conditions for χ and ∫ Ω u 0. This paper derives local existence and extensibility criterion ruling out gradient blow-up when p , q ≥ 1 , and moreover shows global existence and boundedness of solutions when p > q + 1 − 1 n. [ABSTRACT FROM AUTHOR]

The paper focuses on a nonlinear multiparameter eigenvalue problem called P. This problem involves n spectral parameters and also depends on n 2 numerical parameters, where n ⩾ 2 is an integer. If these numerical parameters vanish, P degenerates into n linear (one-parameter) eigenvalue problems called P i 0 (i = 1 , n ‾). In connection with P one can consider another n nonlinear one-parameter eigenvalue problems called P i. The problems P i have eigenvalues with as well as without linear counterparts. The paper suggests to consider P i as 'nonunperturbed' instead of P i 0. Using properties of eigenvalues of P i , one manages to prove existence of eigentuples of P. Among the eigentuples found in this way, there are eigetuples with as well as without linear counterparts. Results of the paper are found with a nonclassical approach. Applications of the found results to nonlinear optics are shown. [ABSTRACT FROM AUTHOR]

Abstract The present paper is devoted to the investigation of the following nonlocal dispersal equation u t (t , x) = D σ m [ ∫ Ω J σ (x − y) u (t , y) d y − u (t , x) ] + f (t , x , u (t , x)) , t > 0 , x ∈ Ω ‾ , where Ω ⊂ R N is a bounded and connected domain with smooth boundary, m ∈ [ 0 , 2) is the cost parameter, D > 0 is the dispersal rate, σ > 0 characterizes the dispersal range, J σ = 1 σ N J (⋅ σ) is the scaled dispersal kernel, and f is a time-periodic nonlinear function of generalized KPP type. This paper is a continuation of the works of Berestycki et al. [3,4] , where f was assumed to be time-independent. We first study the principal spectral theory of the linear operator associated to the linearization of the equation at u ≡ 0. We establish an easily verifiable, general and sharp sufficient condition for the existence of the principal eigenvalue as well as important sup-inf characterizations of the principal eigenvalue. Next, we study the influences of the principal spectrum point on the global dynamics and confirm that the principal spectrum point being zero is critical. It is followed by the investigation of the effects of the dispersal rate D and the dispersal range characterized by σ on the principal spectrum point and the positive time-periodic solution. In particular, we prove various limiting properties of the principal spectrum point and the positive time-periodic solution as D , σ → 0 + or ∞. To achieve these, we develop new techniques to overcome fundamental difficulties caused by the lack of the usual L 2 variational formula for the principal eigenvalue, the lack of the regularizing effects of the semigroup generated by the nonlocal dispersal operator, and the presence of the time-dependence of the nonlinearity f. Finally, we establish the maximum principle for time-periodic nonlocal dispersal operators. [ABSTRACT FROM AUTHOR]

Abstract The Fourier transform, F , on R N (N ≥ 3) transforms the Cauchy problem for the strongly damped wave equation u t t − Δ u t − Δ u = 0 to an ordinary differential equation in time. We let u (t , x) be the solution of the problem given by the Fourier transform, and ν (t , ξ) be the asymptotic profile of F (u) (t , ξ) = u ˆ (t , ξ) found by Ikehata in the paper Asymptotic profiles for wave equations with strong damping (2014). In this paper we study the asymptotic expansions of the squared L 2 -norms of u (t , x) , u ˆ (t , ξ) − ν (t , ξ) , and ν (t , ξ) as t → ∞. With suitable initial data u (0 , x) and u t (0 , x) , we establish the rate of decay of the squared L 2 -norms of u (t , x) and ν (t , ξ) as t → ∞. By noting the cancellation of leading terms of their respective expansions, we conclude that the rate of convergence between u ˆ (t , ξ) and ν (t , ξ) in the L 2 -norm occurs quickly relative to their individual behaviors. This observation is similar to the diffusion phenomenon, which has been well studied. [ABSTRACT FROM AUTHOR]

Abstract In this paper we develop new fundamental results in the Sturmian theory for nonoscillatory linear Hamiltonian systems on an unbounded interval. We introduce a new concept of a multiplicity of a focal point at infinity for conjoined bases and, based on this notion, we prove singular Sturmian separation theorems on an unbounded interval. The main results are formulated in terms of the (minimal) principal solutions at both endpoints of the considered interval, and include exact formulas as well as optimal estimates for the numbers of proper focal points of one or two conjoined bases. As a natural tool we use the comparative index, which was recently implemented into the theory of linear Hamiltonian systems by the authors and independently by J. Elyseeva. Throughout the paper we do not assume any controllability condition on the system. Our results turn out to be new even in the completely controllable case. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we study an implicit version of the sweeping process. Based on methods of convex analysis, we prove the equivalence of the implicit sweeping process with a differential equation, which enables us to show the existence and uniqueness of the solution to the implicit sweeping process in a very general framework. Moreover, this equivalence allows us to give a characterization of nonsmooth Lyapunov pairs and invariance for implicit sweeping processes. The results of the paper are illustrated with two applications to quasistatic evolution variational inequalities and electrical circuits. [ABSTRACT FROM AUTHOR]

Abstract Recently, the authors of [22] studied a diffusive prey–predator model with two different free boundaries. They first obtained the existence, uniqueness, regularity, uniform estimates and long time behaviors of global solution, and then established the conditions for spreading and vanishing. Especially, when spreading occurs, they provided accurate limits of two species as t → + ∞ , and gave some estimates of asymptotic spreading speeds of two species and asymptotic speeds of two free boundaries. Motivated by the paper [22] , in this paper we discuss the diffusive competition model with two different free boundaries, which had been investigated by [7,11,15,21]. The main purpose of this paper is to establish much sharper estimates of asymptotic spreading speeds of two species and asymptotic speeds of two free boundaries when spreading occurs. Furthermore, how the solution approaches the semi-wave when spreading happens is also described. [ABSTRACT FROM AUTHOR]

Abstract This paper considers multidimensional jump type stochastic differential equations with super linear and non-Lipschitz coefficients. After establishing a sufficient condition for nonexplosion, this paper presents sufficient local non-Lipschitz conditions for pathwise uniqueness. The non-confluence property for solutions is investigated. Feller and strong Feller properties under local non-Lipschitz conditions are investigated via the coupling method. Sufficient conditions for irreducibility and exponential ergodicity are derived. As applications, this paper also studies multidimensional stochastic differential equations driven by Lévy processes and presents a Feynman–Kac formula for Lévy type operators. [ABSTRACT FROM AUTHOR]

Abstract The present paper is concerned with the asymptotic behaviors of radially symmetric solutions for the multi-dimensional Burgers equation on the exterior domain in R n , n ≥ 3 , where the boundary and far field conditions are prescribed. We show that in some case where the corresponding 1-D Riemann problem for the non-viscous part admits a shock wave, the solution tends toward a linear superposition of stationary and rarefaction waves as time goes to infinity, and also show the decay rate estimates. Furthermore, we improve the results on the asymptotic stability of the stationary waves which are treated in the previous papers [2] , [3]. Finally, for the case of n = 3 , we give the complete classification of the asymptotic behaviors, which includes even a linear superposition of stationary and viscous shock waves. [ABSTRACT FROM AUTHOR]

Abstract This paper deals with positive solutions of the fully parabolic system { u t = Δ u − χ ∇ ⋅ (u ∇ v) in Ω × (0 , ∞) , τ 1 v t = Δ v − v + w in Ω × (0 , ∞) , τ 2 w t = Δ w − w + u in Ω × (0 , ∞) under mixed boundary conditions (no-flux and Dirichlet conditions) in a smooth bounded convex domain Ω ⊂ R 4 with positive parameters τ 1 , τ 2 , χ > 0 and nonnegative smooth initial data (u 0 , v 0 , w 0). Global existence and boundedness of solutions were shown if ‖ u 0 ‖ L 1 (Ω) < (8 π) 2 / χ in Fujie–Senba (2017). In the present paper, it is shown that there exist blowup solutions satisfying ‖ u 0 ‖ L 1 (Ω) > (8 π) 2 / χ. This result suggests that the system can be regard as a generalization of the Keller–Segel system, which has 8 π / χ -dichotomy. The key ingredients are a Lyapunov functional and quantization properties of stationary solutions of the system in R 4. [ABSTRACT FROM AUTHOR]

Abstract We study the non-linear minimization problem on H 0 1 (Ω) ⊂ L q with q = 2 n n − 2 , α > 0 and n ≥ 4 : inf u ∈ H 0 1 (Ω) ‖ u ‖ L q = 1 ⁡ ∫ Ω a (x , u) | ∇ u | 2 − λ ∫ Ω | u | 2 where a (x , s) presents a global minimum α at (x 0 , 0) with x 0 ∈ Ω. In order to describe the concentration of u (x) around x 0 , one needs to calibrate the behavior of a (x , s) with respect to s. The model case is inf u ∈ H 0 1 (Ω) ‖ u ‖ L q = 1 ⁡ ∫ Ω (α + | x | β | u | k) | ∇ u | 2 − λ ∫ Ω | u | 2. In a previous paper dedicated to the same problem with λ = 0 , we showed that minimizers exist only in the range β < k n / q , which corresponds to a dominant non-linear term. On the contrary, the linear influence for β ≥ k n / q prevented their existence. The goal of this present paper is to show that for 0 < λ ≤ α λ 1 (Ω) , 0 ≤ k ≤ q − 2 and β > k n / q + 2 , minimizers do exist. [ABSTRACT FROM AUTHOR]

Abstract This is a short note to complete the paper appeared in Francini et al. (2016) [4] , where a rough version of the classical well known Hadamard three-circle theorem for solution of an elliptic PDE in divergence form has been proved. Precisely, instead of circles, the authors obtain a similar inequality in a more complicated geometry. In this paper we clean the geometry and obtain a generalized version of the three-circle inequality for elliptic equation with coefficients with discontinuity of jump type. [ABSTRACT FROM AUTHOR]

Abstract Spatially localized patterns have been observed in numerous physical contexts, and their bifurcation diagrams often exhibit similar snaking behavior: symmetric solution branches, connected by bifurcating asymmetric solution branches, wind back and forth in an appropriate parameter. Previous papers have addressed existence of such solutions; here we address their stability, taking the necessary first step of unifying existence and uniqueness proofs for symmetric and asymmetric solutions. We then show that, under appropriate assumptions, temporal eigenvalues of the front and back underlying a localized solution are added with multiplicity in the right half plane. In a companion paper, we analyze the behavior of eigenvalues at λ = 0 and inside the essential spectrum. Our results show that localized snaking solutions are stable if, and only if, the underlying fronts and backs are stable: unlike localized non-oscillatory solutions, no interaction eigenvalues are present. We use the planar Swift–Hohenberg system to illustrate our results. [ABSTRACT FROM AUTHOR]

Abstract There has been a substantial amount of well mixing epidemic models devoted to characterizing the observed complex phenomena (such as bistability, hysteresis, oscillations, etc.) during the transmission of many infectious diseases. A comprehensive explanation of these phenomena by epidemic models on complex networks is still lacking. In this paper we study epidemic dynamics in an adaptive network proposed by Gross et al., where the susceptibles are able to avoid contact with the infectious by rewiring their network connections. Such rewiring of the local connections changes the topology of the network, and inevitably has a profound effect on the transmission of the disease, which in turn influences the rewiring process. We rigorously prove that the adaptive epidemic model investigated in this paper exhibits degenerate Hopf bifurcation, homoclinic bifurcation and Bogdanov–Takens bifurcation. Our study shows that adaptive behaviors during an epidemic may induce complex dynamics of disease transmission, including bistability, transient and sustained oscillations, which contrast sharply to the dynamics of classical network models. Our results yield deeper insights into the interplay between topology of networks and the dynamics of disease transmission on networks. [ABSTRACT FROM AUTHOR]

Abstract This paper considers the inverse problem of scattering of time-harmonic acoustic and electromagnetic plane waves by a bounded, inhomogeneous, penetrable obstacle with embedded objects inside. A new method is proposed to prove that the inhomogeneous penetrable obstacle can be uniquely determined from the far-field pattern at a fixed frequency, disregarding its contents. Our method is based on constructing a well-posed interior transmission problem in a small domain associated with the Helmholtz or modified Helmholtz equation and the Maxwell or modified Maxwell equations. A key role is played by the smallness of the domain which ensures that the lowest transmission eigenvalue is large so that a given wave number k is not an eigenvalue of the interior transmission problem. Another ingredient in our proofs is a priori estimates of solutions to the transmission scattering problems with data in L p (1 < p < 2), which are established in this paper by using the integral equation method. A main feature of the new method is that it can deal with the acoustic and electromagnetic cases in a unified way and can be easily applied to deal with inverse scattering by unbounded rough interfaces. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the problem { u t t − Δ u + P ( x , u t ) = f ( x , u ) in ( 0 , ∞ ) × Ω , u = 0 on ( 0 , ∞ ) × Γ 0 , u t t + ∂ ν u − Δ Γ u + Q ( x , u t ) = g ( x , u ) on ( 0 , ∞ ) × Γ 1 , u ( 0 , x ) = u 0 ( x ) , u t ( 0 , x ) = u 1 ( x ) in Ω ‾ , where Ω is a bounded open subset of R N with C 1 boundary ( N ≥ 2 ), Γ = ∂ Ω , Γ 1 is relatively open on Γ, Δ Γ denotes the Laplace–Beltrami operator on Γ, ν is the outward normal to Ω, and the terms P and Q represent nonlinear damping terms, while f and g are nonlinear perturbations. In the paper we establish local and global existence, uniqueness and Hadamard well-posedness results when source terms can be supercritical or super-supercritical. [ABSTRACT FROM AUTHOR]

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 This paper deals with a fourth order parabolic PDE arising in the theory of epitaxial growth of crystal. We focalize the study on one open question proposed by Escudero et al. (2015) [4] , that is, L p -norm blow-up. For the initial energy J (u 0) < λ 1 6 ‖ u 0 ‖ 2 2 , we prove the solution blows up in finite time with L 2 -norm, where λ 1 is the least Dirichlet eigenvalue of the biharmonic operator. Moreover, the lifespan of the solution is got, and the results of this paper also generalize the results got by Xu and Zhou (2017) [12]. [ABSTRACT FROM AUTHOR]

Abstract This paper is concerned with time-delayed reaction–diffusion equations with degenerate diffusion. When the term for birth rate is a nonlocal integral with a heat kernel, the family of minimum wave speeds corresponding to all the degenerate diffusion coefficients is proved to admit a uniform positive infimum. However, when the term for birth rate is local, there is no positive infimum of all the minimum wave speeds. This difference indicates that the nonlocal effect plays a role as Laplacian such that a positive lower bound independent of the degenerate diffusion exists for the minimum wave speeds. The approach adopted for the proof is the monotone technique with the viscosity vanishing method. The degeneracy of diffusion for the equation causes us essential difficulty in the proof. A number of numerical simulations are also carried out at the end of the paper, which further numerically confirm our theoretical results. [ABSTRACT FROM AUTHOR]