The aim of this paper is to study the stochastic SIR equation with general incidence functional responses and in which both natural death rates and the incidence rate are perturbed by white noises. We derive a sufficient and almost necessary condition for the extinction and permanence for SIR epidemic system with multi noises{dS(t)=[a1−b1S(t)−I(t)f(S(t),I(t))]dt+σ1S(t)dB1(t)−I(t)g(S(t),I(t))dB3(t)

Starting from an age-structured diffusive population growth law for single species in a discrete and periodic habitat, we formulate a stage structured population model with spatially periodic dispersal, mortality and recruitment. With a KPP type setting, after establishing the fundamental solution of a discretized heat equation with spatially periodic dispersal, we apply some recently developed dynamical

Maximum Principles on unbounded domains play a crucial rôle in several problems related to linear second-order PDEs of elliptic and parabolic type. In this paper we consider a class of sub-elliptic operators L in RN and we establish some criteria for an unbounded open set to be a Maximum Principle set for L. We extend some classical results related to the Laplacian (by Deny, Hayman and Kennedy) and

In this work we prove local existence of strong solutions to the initial-value problem arising in one-dimensional strain-limiting viscoelasticity, which is based on a nonlinear constitutive relation between the linearized strain, the rate of change of the linearized strain and the stress. The model is a generalization of the nonlinear Kelvin-Voigt viscoelastic solid under the assumption that the strain

We consider the initial value problem (IVP) for the fractional Korteweg-de Vries equation (fKdV)(0.1){∂tu−Dxα∂xu+u∂xu=0,x,t∈R,0<α<1,u(x,0)=u0(x). It has been shown that the solutions to certain dispersive equations satisfy the propagation of regularity phenomena. More precisely, it deals in determine whether regularity of the initial data on the right hand side of the real line is propagated to the

The global boundedness and the hair trigger effect of solutions for the nonlinear nonlocal reaction-diffusion equation∂u∂t=Δu+μuα(1−κJ⁎uβ),inRN×(0,∞),N≥1 with α≥1, β,μ,κ>0 and u(x,0)=u0(x) are investigated. Under appropriate assumptions on J, it is proved that for any nonnegative and bounded initial condition, if α∈[1,α⁎) with α⁎=1+β for N=1,2 and α⁎=1+2βN for N>2, then the problem has a global bounded

In this paper, we consider several geometric inverse problems for linear elliptic systems. We prove uniqueness and stability results. In particular, we show the way that the observation depends on the perturbations of the domain. In some particular situations, this provides a strategy that could be used to compute approximations to the solution of the inverse problem. In the proofs, we use techniques

This paper is concerned with the following planar Schrödinger-Poisson system{−Δu+V(x)u+ϕu=f(x,u),x∈R2,Δϕ=u2,x∈R2, where V∈C(R2,[0,∞)) is axially symmetric and f∈C(R2×R,R) is of subcritical or critical exponential growth in the sense of Trudinger-Moser. We obtain the existence of a nontrivial solution or a ground state solution of Nehari-type and infinitely many solutions to the above system under weak

Boundary layer behaviour of a family of second order nonlinear differential equations with Neumann boundary condition arising from an order-reduction of a pseudo-differential equation in fluid dynamics is analysed. The problem is considered with two perturbation parameters μ,δ. Using asymptotic and numerical methods it is shown that by perturbing δ the problem in the limit μ→0 changes from a homogeneous

In this work we prove the existence of standing-wave solutions to the scalar non-linear Klein-Gordon equation in dimension one and the stability of the ground-state, the set which contains all the minima of the energy constrained to the manifold of the states sharing a fixed charge. For non-linearities which are combinations of two competing powers we prove that standing-waves in the ground-state are

In this paper, we study the cyclicity of periodic annulus and Hopf cyclicity in perturbing a quintic Hamiltonian system. The undamped system is hyper-elliptic, non-symmetric with a degenerate heteroclinic loop, which connects a hyperbolic saddle to a nilpotent saddle. We rigorously prove that the cyclicity is 3 for periodic annulus when the weak damping term has the same degree as that of the associated

This paper is concerned with the global well-posedness issue of the compressible viscoelastic fluids in the whole space RN,N≥2. The proof relies on the dispersive estimates to the linearized hyperbolic system combined with energy estimates to the hyperbolic-parabolic system. By exploiting the intrinsic structure of the system under some physical restrictions, we find that the compressible viscoelastic

For the Cauchy problem of multidimensional quasilinear hyperbolic systems of diagonal form without self-interaction, we show the global existence of classical solutions with small initial data. We also prove that the global solution will converge to a solution of the linearized system as time tends to infinity, and the convergence rate is also determined.

Models issued from ecology, chemical reactions and several other application fields lead to semi-linear parabolic equations with super-linear growth. Even if, in general, blow-up can occur, these models share the property that mass control is essential. In many circumstances, it is known that this L1 control is enough to prove the global existence of weak solutions. The theory is based on basic estimates

In this paper, we consider three-dimensional equatorial flows with constant vorticity beneath a wave train and above a flat bed in the β-plane approximation with centripetal forces. As a striking upshot, we show that the equatorial flow is necessarily irrotational and the free surface is necessarily flat if it exhibits a constant vorticity, owing to the presence of centripetal forces. For the case

We consider transmission problems for a coupling of a string and a beam with at least one of them being thermoelastic. The heat conduction is modeled by Fourier's law. We prove that the associated semigroup is exponentially stable when the string is thermoelastic. If only the beam is thermoelastic the system is always polynomially stable, but non exponentially stable for a large class of beams.

In this paper, we derive sharp bounds on the semigroup of the linearized incompressible Navier-Stokes equations near a stationary shear layer in the half space (R+2 or R+3), with Dirichlet boundary conditions, assuming that this shear layer in spectrally unstable for Euler equations. In the inviscid limit, due to the prescribed no-slip boundary conditions, vorticity becomes unbounded near the boundary

Let G be an infinite discrete countable amenable group acting continuously on a compact metrizable space X and Ω be a sequence in G. By using open covers of X, a topological invariant, the topological sequence entropy, is defined. It is shown that the topological sequence entropy can also be defined through relative spanning set or relative separate set. We prove that the topological sequence entropy

We first present a warped product manifold with boundary to show the non-uniqueness of the positive constant scalar curvature and positive constant boundary mean curvature equation. Next, we construct a smooth counterexample to show that the compactness of the set of “lower energy” solutions to the above equation fails when the dimension of the manifold is not less than 62.

The Landau–Lifshitz–Bloch equation perturbed by a space-dependent noise was proposed in [10] as a model for evolution of spins in ferromagnetic materials at the full range of temperatures, including the temperatures higher than the Curie temperature. In the case of a ferromagnet filling a bounded domain D⊂Rd, d=1,2,3, we show the existence of strong (in the sense of PDEs) martingale solutions. Furthermore

We consider travelling times of billiard trajectories in the exterior of an obstacle K on a two-dimensional Riemannian manifold M. We prove that given two obstacles with almost the same travelling times, the generalised geodesic flows on the non-trapping parts of their respective phase-spaces will have a time-preserving conjugacy. Moreover, if M has non-positive sectional curvature, we prove that if

For an optimal control problem governed by a controlled nonconvex sweeping process, we provide, using an exponential penalization technique, existence of solution and nonsmooth necessary conditions in the form of the Pontryagin maximum principle. Our results generalize known theorems in the literature, including those in [3] and [23], in several directions. Indeed, the main feature in our sweeping

The article is intended to study the Hopf−Zero bifurcation of a novel age-dependent predator−prey system with Monod−Haldane functional response comprising strong Allee effect. Here in this system the predators fertility function f(x) is regarded as a piecewise function with respect to their maturation period τ. The system is interpreted as a non-densely defined abstract Cauchy problem, and the condition

In this paper, complete integrability and linearizability of cubic Z2 systems with two non-resonant and elementary singular points are investigated. First of all, four simple normal forms are obtained based on the coefficients and eigenvalues of cubic Z2 systems. Then, the integrable and linearizable conditions are classified according to the four different cases respectively, and the problem is solved

We study existence of global solutions and finite time blow-up of solutions to the Cauchy problem for the porous medium equation with a variable density ρ(x) and a power-like reaction term ρ(x)up with p>1; this is a mathematical model of a thermal evolution of a heated plasma (see [29]). The density decays slowly at infinity, in the sense that ρ(x)≲|x|−q as |x|→+∞ with q∈[0,2). We show that for large

We consider the singular limit problem of the Cauchy problem to the Keller-Segel equation in the two dimensional critical space. It is shown that the solution to the Keller-Segel system in the scaling critical function space converges to the solution to the drift-diffusion system of parabolic-elliptic equations (the simplified Keller-Segel equation) in the critical space strongly as the relaxation

The paper studies existence of solutions for the nonlinear Schrödinger equation(0.1)−(∇+iA(x))2u+V(x)u=f(|u|)u with a general bounded external magnetic field. In particular, no lattice periodicity of the magnetic field or presence of external electric field is required. Solutions are obtained by means of a general structural statement about bounded sequences in the magnetic Sobolev space.

This paper develops a well-posedness theory for hyperbolic Maxwell obstacle problems generalizing the result by Duvaut and Lions (1976) [5]. Building on the recently developed result by Yousept (2020) [30], we prove an existence result and study the uniqueness through a local H(curl)-regularity analysis with respect to the constraint set. More precisely, every solution is shown to locally satisfy the

We consider a population dynamics model with degenerate diffusion and time delay. We discover sharp-oscillatory waves with sharp edges and non-decaying oscillations arising from density-dependent dispersal and reproduction with age structure. Degenerate diffusion and bad effect of time delay prevent the use of existing approaches. Here, we develop a new delayed iteration framework to show the existence

In the paper, we consider the initial value problem to the Camassa-Holm equation in the real-line case. Based on the local well-posedness result and the lifespan, we proved that the data-to-solution map of this problem is not uniformly continuous in nonhomogeneous Besov spaces in the sense of Hadamard. This result improves considerably the previous work given by Himonas–Kenig [23].

We study the well-posedness and regularity theory for the Radiative Transfer equation in the peaked regime posed in the half-space. An average lemma for the transport equation in the half-space is established and used to generate interior regularity for solutions of the model. The averaging also shows a fractional regularisation gain up to the boundary for the spatial derivatives.

For models describing water waves, Constantin and Escher [3]'s works have long been considered as the cornerstone method for proving wave breaking phenomena. Their rigorous analytic proof shows that if the lowest slope of flows can be controlled by its highest slope initially, then the wave-breaking occur for the Whitham-type equation. Since this breakthrough, there have been numerous refined wave-breaking

The primary goal of this paper is to understand how higher-order nonlinearities affect the dispersive dynamics. As a prototype, we study a class of higher-order Camassa–Holm equations which can be viewed as a generalization of the Camassa–Holm equation. We develop a delicate analysis to investigate the formation of singularities and provide sufficient conditions on initial data that lead to the finite

We consider a viscoelastic plate equation of Moore-Gibson-Thompson type coupled with two different kinds of thermal laws, namely, the usual Fourier one and the heat conduction law of type III. In both cases, the resulting system is shown to generate a contraction semigroup of solutions on a suitable Hilbert space. Then we prove that these semigroups are analytic, despite the fact that the semigroup

In this paper we continue the study of non-diagonalisable hyperbolic systems with variable multiplicity started by the authors in [1]. In the case of space dependent coefficients, we prove a representation formula for solutions that allows us to derive results of regularity and propagation of singularities.

The generalized Huygens' principle for the non-isentropic bipolar Navier-Stokes-Poisson system in dimension three is given, which is different from the unipolar case. All of the previous estimates containing Huygens' waves on wave behaviors for compressible flow models rely heavily on the conservative structure, for instance, the Navier-Stokes system and even the isentropic bipolar Navier-Stokes-Poisson

The abstract theory of boundary triples is applied to the classical Jacobi differential operator and its powers in order to obtain the Weyl m-function for several self-adjoint extensions with interesting boundary conditions: separated, periodic and those that yield the Friedrichs extension. These matrix-valued Nevanlinna–Herglotz m-functions are, to the best knowledge of the author, the first explicit

A conjecture on the nature of the simultaneous binary collision is explored in the collinear 4-body problem. It is known that any attempt to remove the singularity via block regularisation will result in a regularised flow that is no more than C8/3 differentiable with respect to initial conditions. Through a blow-up of the singularity, this loss of differentiability is investigated and a new proof

We investigate the quasi-neutral limit (the zero Debye length limit) for the Euler-Poisson system with radial symmetry in an annular domain. Under physically relevant conditions at the boundary, referred to as the Bohm criterion, we first construct the approximate solutions by the method of asymptotic expansion in the limit parameter, the square of the rescaled Debye length, whose detailed derivation

In this paper, we study the free boundary problem of compressible non-isothermal liquid crystal flow in dimension one. For the vacuum interface condition, that is, the density is assumed to vanish and connect to vacuum continuously, we prove global existence and uniqueness of classical solutions to the liquid crystal system with large initial data.

This paper is concerned with the asymptotic stability analysis of a one dimensional wave equation with Dirichlet boundary conditions subject to a nonlinear distributed damping within an Lp functional framework, p∈[2,∞]. Some well-posedness results are provided together with exponential decay to zero of trajectories, with an estimation of the decay rate. The well-posedness results are proved by considering

The global-in-time solutions with discontinuous initial data, when the density has no regularity, are constructed in [10], [11], [12] for the isentropic compressible Navier-Stokes equations in multi-dimensional spaces. The time decay rates of these solutions with low regularity still remain unsolved. In this paper we establish the decay rates of solutions in [10], [11], [12] in Lr-norm with 2≤r≤∞ and

As shown in [17], [18], for the hydrodynamic model of semiconductors represented by Euler-Poisson equations with sonic boundary and subsonic/supersonic doping profile, the structure of stationary solutions are very complicated. It may possess various solutions like subsonic/supersonic/transonic flows. In this paper, we consider a more challenging case where the doping profile is transonic, which is

In this paper we prove stability results for a semilinear hyperbolic coupled system subject to a viscoelastic localized damping acting in the first equation and a frictional localized one acting in the second equation of the system. We divide the proof into two parts. In the first part the equations are posed in a homogeneous medium Ω with the damping acting in a boundary neighbourhood. In the second

We study the so-called Hyperbolic Mean Curvature (HMC) flow introduced by LeFloch and Smoczyk in 2008 for the evolution of a closed hypersurface moving in the direction of its mean curvature vector. This flow stems from a geometrically natural action consisting of a kinetic energy and an internal energy. We study the initial value problem for this flow in the case of an entire graph (in arbitrary dimension)

We establish the higher differentiability of solutions to a class of obstacle problems of the typemin⁡{∫Ωf(x,Dv(x))dx:v∈Kψ(Ω)}, where ψ is a fixed function called obstacle, Kψ(Ω)={v∈Wloc1,p(Ω,R):v≥ψ a.e. in Ω} and the convex integrand f satisfies p-growth conditions with respect to the gradient variable. We derive that the higher differentiability property of the weak solution v is related to the regularity

The boundary value problem for the three-dimensional steady viscous compressible Navier-Stokes/Allen-Cahn (NSAC) system is considered. We establish the existence of a weak solution for the adiabatic index γ>2 with the phase function belonging to [−1,1]. Moreover, we give further analysis on the weak solution. More specifically, we prove that the steady NSAC system degenerates into Navier-Stokes system

In this paper we study the Type IIb mean curvature flow. We first prove that if the convex entire graph (y,u(|y|)) over Rn, n≥2, satisfying there exist positive constants ϵ, c and N such that u′(r)≥crϵ for r≥N, the longtime solution to mean curvature flow with initial data (y,u(|y|)) must be Type IIb. We also study the asymptotic behavior of Type IIb mean curvature flow and show that the limit of suitable

In this paper, the dynamics of a general predator-prey system with diffusion and stage structures are investigated. Under some assumptions, the stability of positive steady state and the existence of local Hopf bifurcation are obtained by analyzing the distribution of eigenvalues. Moreover, the global existence of positive periodic solution is established by using the global Hopf bifurcation result

In this paper, we study the initial value problem for semilinear wave equations with the time-dependent and scale-invariant damping in two dimensions. Similarly to the one dimensional case by Kato, Takamura and Wakasa in 2019, we obtain the lifespan estimates of the solution for a special constant in the damping term, which are classified by total integral of the sum of the initial position and speed

In this paper we study unfoldings of planar vector fields in a neighbourhood of a hyperbolic resonant saddle. We give a structure theorem for the asymptotic expansion of the local Dulac time (as well as the local Dulac map) with the remainder uniformly flat with respect to the unfolding parameters. Here local means close enough to the saddle in order that the normalizing coordinates provided by a suitable

This paper concerns an initial value problem of compressible Navier-Stokes-Poisson equations subject to large and non-flat doping profile in whole space R3. The global well-posedness of strong solutions with large oscillations and vacuum is established, provided the initial data are of small energy and the steady state is strictly away from vacuum. A weak-strong uniqueness result is also obtained.

We propose a general framework for proving that a compact, infinite-dimensional map has an invariant set on which the dynamics is semiconjugated to a subshift of finite type. The method is then applied to certain Poincaré map of the Kuramoto-Sivashinsky PDE on the line with odd and periodic boundary conditions and with parameter ν=0.1212. We give a computer-assisted proof of the existence of symbolic

In this paper, we prove a variational formula of torsional rigidity under the Orlicz sum, and then introduce the Orlicz mixed torsional rigidity which yields the Orlicz-Brunn-Minkowski inequality and Orlicz-Minkowski inequality for torsional rigidity. As an application, we prove the existence of solution to the Orlicz-Minkowski problem for discrete measure and general measure associated with torsional

In this work, we investigate the problem of finite time blow up as well as the upper bound estimates of lifespan for solutions to small-amplitude semilinear wave equations with mixed nonlinearities c1|ut|p+c2|u|q, posed on asymptotically Euclidean manifolds, which is related to both the Strauss conjecture and the Glassey conjecture. In some cases, we obtain existence results, where the lower bound

An infinite lattice model of a recurrent neural network with random connection strengths between neurons is developed and analyzed. To incorporate the presence of various type of delays in the neural networks, both discrete and distributed time varying delays are considered in the model. For the existence of random pullback attractors and periodic attractors, the nonlinear terms of the resulting system

We prove the existence of a relative finite-energy solution of the one-dimensional, isentropic Euler equations under the assumption of an asymptotically isothermal pressure law, that is, p(ρ)/ρ=O(1) in the limit ρ→∞. This solution is obtained as the vanishing viscosity limit of classical solutions of the one-dimensional, isentropic, compressible Navier–Stokes equations. Our approach relies on the method

We study the approximate controllability of a general class of first order abstract control differential problems with state-dependent delay. To this end, we establish and prove several new results on local and global existence and uniqueness of mild and strict solutions for abstract differential equations with state-dependent delay. In the last section, some examples are presented.

We study a three-dimensional fluid model describing rapidly rotating convection that takes place in tall columnar structures. The purpose of this model is to investigate the cyclonic and anticyclonic coherent structures. Global existence, uniqueness, continuous dependence on initial data, and large-time behavior of strong solutions are shown provided the model is regularized by a weak dissipation term

In this paper, we consider eigenvalue problems of a second-order measure differential equation. We first study some general theories regarding the completely continuity of eigenvalues, the variational characterizations of eigenvalues, and the oscillating properties of eigenfunctions. Then, we solve the following minimization problem: when the m-th Neumann eigenvalue is given, to find explicitly what