American Institute of Mathematical Sciences

This expository paper examines key results on the dynamics of nonlinear conservation laws with random initial data and applies some theorems to physically important situations. Conservation laws with some nonlinearity, e.g. Burgers' equation, exhibit discontinuous behavior, or shocks, even for smooth initial data. The introduction of randomness in any of several forms into the initial condition renders the analysis extremely complex. Standard methods for tracking a multitude of shock collisions are difficult to implement, suggesting other methods may be needed. We review several perspectives into obtaining the statistics of resulting states and shocks. We present a spectrum of results from a number of works, both deterministic and random. Some of the deep theorems are applied to important discrete examples where the results can be understood in a clearer, more physical context.

This paper mainly aims to study the influence of individuals' different heterogeneous contact patterns on the spread of the disease. For this purpose, an SIS epidemic model with a general form of heterogeneous infection rate is investigated on complex heterogeneous networks. A qualitative analysis of this model reveals that, depending on the epidemic threshold \begin{document}$R_0$ \end{document}, either the disease-free equilibrium or the endemic equilibrium is globally asymptotically stable. Interestingly, no matter what functional form the heterogeneous infection rate is, whether the disease will disappear or not is completely determined by the value of \begin{document}$R_0$ \end{document}, but the heterogeneous infection rate has close relation with the epidemic threshold \begin{document}$R_0$ \end{document}. Especially, the heterogeneous infection rate can directly affect the final number of infected nodes when the disease is endemic. The obtained results improve and generalize some known results. Finally, based on the heterogeneity of contact patterns, the effects of different immunization schemes are discussed and compared. Meanwhile, we explore the relation between the immunization rate and the recovery rate, which are the two important parameters that can be improved. To illustrate our theoretical results, the corresponding numerical simulations are also included.

This paper is concerned with a class of advection hyperbolic-parabolic systems with nonlocal delay. We prove that the wave profile is described by a hybrid system that consists of an integral transformation and an ordinary differential equation. By considering the same problem for a properly parameterized system and the continuous dependence of the wave speed on the parameter involved, we obtain the existence and uniqueness of traveling wave solutions in advection hyperbolic-parabolic system with nonlocal delay under bistable assumption. The influence of advection on the propagation speed is also considered.

In this paper, a Lévy-diffusion Leslie-Gower predator-prey model with a nonmonotonic functional response is studied. We show the existence, uniqueness and attractiveness of the globally positive solution to this model. Moreover, to its corresponding steady-state model, we obtain the stability of the semi-trivial solutions, the existence and nonexistence of coexistence states by the method of topological degree, the uniqueness and stability of coexistence state, and the multiplicity and stability of coexistence states by Grandall-Rabinowitz bifurcation theorem. In addition, to get these results, we study the property of the Lévy diffusion operator, and give out the comparison principle of the generalized parabolic Lévy-diffusion differential equation, as well as the existence and stability of the solution for the steady-state Logistic equation with Lévy diffusion. Furthermore, we obtain the comparison principle of the steady-state Lévy-diffusion equation. As far as we know, these results are new in the ecological model.

A Markov-modulated framework is presented to incorporate correlated inter-event times into the stochastic susceptible-infectious-recovered (SIR) epidemic model for a closed finite community. The resulting process allows us to deal with non-exponential distributional assumptions on the contact process between the compartment of infectives and the compartment of susceptible individuals, and the recovery process of infected individuals, but keeping the dimensionality of the underlying Markov chain model tractable. The variability between SIR-models with distinct level of correlation is discussed in terms of extinction times, the final size of the epidemic, and the basic reproduction number, which is defined here as a random variable rather than an expected value.

In this paper, we consider numerical approximations for a model of smectic-A liquid crystal flows in its weak flow limit. The model, derived from the variational approach of the de Gennes free energy, is consisted of a highly nonlinear system that couples the incompressible Navier-Stokes equations with two nonlinear order parameter equations. Based on some subtle explicit-implicit treatments for nonlinear terms, we develop an unconditionally energy stable, linear and decoupled time marching numerical scheme for the reduced model in the weak flow limit. We also rigorously prove that the numerical scheme obeys the energy dissipation law at the discrete level. Various numerical simulations are presented to demonstrate the accuracy and the stability of the scheme.

This paper studies the pullback asymptotic behavior of solutions for a non-autonomous incompressible non-Newtonian fluid on 2D bounded domains. We show existence of the pullback exponential attractor introduced by Langa, Miranville and Real [27], moreover, give existence of the global pullback attractor with finite fractal dimension and reveal the relationship between the global pullback attractor and the pullback exponential attractor. These results improve our previous associated results in papers [29,40] for the non-Newtonian fluid.

The mild Itô formula proposed in Theorem 1 in [Da Prato, G., Jentzen, A., & Röckner, M., A mild Ito formula for SPDEs, arXiv: 1009.3526 (2012), To appear in the Trans. Amer. Math. Soc.] has turned out to be a useful instrument to study solutions and numerical approximations of stochastic partial differential equations (SPDEs) which are formulated as stochastic evolution equations (SEEs) on Hilbert spaces. In this article we generalize this mild Itô formula so that it is applicable to stopping times instead of deterministic time points and so that it is applicable to solutions and numerical approximations of SPDEs which are formulated as SEEs on UMD (unconditional martingale differences) Banach spaces. These generalizations are especially useful for proving essentially sharp weak convergence rates for numerical approximations of SPDEs such as stochastic heat equations with nonlinear diffusion coefficients.

We are concerned with a two-component reaction-advection-diffusion Lotka-Volterra competition system with constant diffusion rates subject to homogeneous Neumann boundary conditions. We first prove the global existence and uniform boundedness of positive classical solutions to this system. This result complements some of the global existence results in [Y. Lou, M. Winkler and Y. Tao, SIAM J. Math. Anal., 46 (2014), 1228-1262.], where one diffusion rate is taken to be a linear function of the population density. Our second result proves that the total population of each species admits a positive lower bound, under some conditions of system parameters (e.g., when the intraspecific competition rates are large). This result of population persistence indicates that the two competing species coexist over the habitat in a long time.

We study analytically and numerically stability and interaction patterns of quantized vortex lattices governed by the reduced dynamical lawa system of ordinary differential equations (ODEs) - in superconductivity. By deriving several non-autonomous first integrals of the ODEs, we obtain qualitatively dynamical properties of a cluster of quantized vortices, including global existence, finite time collision, equilibrium solution and invariant solution manifolds. For a vortex lattice with 3 vortices, we establish orbital stability when they have the same winding number and find different collision patterns when they have different winding numbers. In addition, under several special initial setups, we can obtain analytical solutions for the nonlinear ODEs.

We analytically study the Hamiltonian system in \begin{document}$ \mathbb{R}^6$\end{document} with Hamiltonian

\begin{document}$H = 1/2 (p_x^2+p_y^2+p_z^2)+\frac{1}{2} (ω_1^2 x^2+ω_2^2 y^2+ ω_3^2 z^2)+ \varepsilon(a z^3 + z (b x^2 +c y^2)),$ \end{document}

being \begin{document}$ a,b,c∈\mathbb{R}$\end{document} with \begin{document}$ c\ne 0$\end{document}, \begin{document}$ \varepsilon$\end{document} a small parameter, and \begin{document}$ ω_1$\end{document}, \begin{document}$ ω_2$\end{document} and \begin{document}$ ω_3$\end{document}the unperturbed frequencies of the oscillations along the \begin{document}$ x$\end{document}, \begin{document}$ y$\end{document} and \begin{document}$ z$\end{document} axis, respectively. For \begin{document}$ |\varepsilon|>0$\end{document} small, using averaging theory of first and second order we find periodic orbits in every positive energy level of \begin{document}$ H$\end{document} whose frequencies are \begin{document}$ ω_1 = ω_2 = ω_3/2$\end{document} and \begin{document}$ ω_1 = ω_2 = ω_3$\end{document}, respectively (the number of such periodic orbits depends on the values of the parameters \begin{document}$ a,b,c$\end{document}). We also provide the shape of the periodic orbits and their linear stability.

We propose and analyze a multiscale model for acid-mediated tumor invasion accounting for stochastic effects on the subcellular level. The setting involves a PDE of reaction-diffusion-taxis type describing the evolution of the tumor cell density, the movement being directed towards pH gradients in the local microenvironment, which is coupled to a PDE-SDE system characterizing the dynamics of extracellular and intracellular proton concentrations, respectively. The global well-posedness of the model is shown and numerical simulations are performed in order to illustrate the solution behavior.

This study considers the problem of the extreme behavior exhibited by solutions to Burgers equation subject to stochastic forcing. More specifically, we are interested in the maximum growth achieved by the "enstrophy" (the Sobolev \begin{document}$H^1$\end{document} seminorm of the solution) as a function of the initial enstrophy \begin{document}$\mathcal{E}_0$\end{document}, in particular, whether in the stochastic setting this growth is different than in the deterministic case considered by Ayala & Protas (2011). This problem is motivated by questions about the effect of noise on the possible singularity formation in hydrodynamic models. The main quantities of interest in the stochastic problem are the expected value of the enstrophy and the enstrophy of the expected value of the solution. The stochastic Burgers equation is solved numerically with a Monte Carlo sampling approach. By studying solutions obtained for a range of optimal initial data and different noise magnitudes, we reveal different solution behaviors and it is demonstrated that the two quantities always bracket the enstrophy of the deterministic solution. The key finding is that the expected values of the enstrophy exhibit the same power-law dependence on the initial enstrophy \begin{document}$\mathcal{E}_0$\end{document} as reported in the deterministic case. This indicates that the stochastic excitation does not increase the extreme enstrophy growth beyond what is already observed in the deterministic case.

A mathematical model, based on a mesoscopic approach, describing the competition between tumor cells and immune system in terms of kinetic integro-differential equations is presented. Four interacting components are considered, representing, respectively, tumors cells, cells of the host environment, cells of the immune system, and interleukins, which are capable to modify the tumor-immune system interaction and to contribute to destroy tumor cells. The internal state variable (activity) measures the capability of a cell of prevailing in a binary interaction. Under suitable assumptions, a closed set of autonomous ordinary differential equations is then derived by a moment procedure and two three-dimensional reduced systems are obtained in some partial quasi-steady state approximations. Their qualitative analysis is finally performed, with particular attention to equilibria and their stability, bifurcations, and their meaning. Results are obtained on asymptotically autonomous dynamical systems, and also on the occurrence of a particular backward bifurcation.

Some diseases such as herpes, bovine and human tuberculosis exhibit relapse in which the recovered individuals do not acquit permanent immunity but return to infectious class. Such diseases are modeled by SIRI models. In this paper, we establish the existence of a unique global positive solution for a stochastic epidemic model with relapse and jumps. We also investigate the dynamic properties of the solution around both disease-free and endemic equilibria points of the deterministic model. Furthermore, we present some numerical results to support the theoretical work.

In this paper, we consider long time behavior of the Cox-Ingersoll-Ross (CIR) interest rate model driven by stable processes with Markov switching. Under some assumptions, we prove an ergodicity-transience dichotomy, namely, the interest rate process is either ergodic or transient. The sufficient and necessary conditions for ergodicity and transience of such interest model are given under some assumptions. Finally, an application to interval estimation of the interest rate processes is presented to illustrate our results.

We consider the perturbed dynamical system applied to non expanding piecewise linear maps on \begin{document}$[0, 1]$\end{document} which describe simplified dynamics of a single neuron. It is known that the Markov operator generated by this perturbed system has asymptotic periodicity with period \begin{document}$n≥1$\end{document}. In this paper, we give a sufficient condition for \begin{document}$n>1$\end{document}, asymptotic periodicity, and for \begin{document}$n = 1$\end{document}, asymptotic stability. That is, we show that there exists a threshold of noises \begin{document}$θ_{*}$\end{document} such that the Markov operator generated by this perturbed system displays asymptotic periodicity (asymptotic stability) if a maximum value of noises is less (greater) than \begin{document}$θ_{*}$\end{document}. This result indicates that an existence of phenomenon called mode-locking is mathematically clarified for this perturbed system.

We prove that for a quadratic polynomial differential system having three pairs of diametrally opposite equilibrium points at infinity that are positively rationally independent, has at most one algebraic limit cycle. Our result provides a partial positive answer to the following conjecture: Quadratic polynomial differential systems have at most one algebraic limit cycle.

We propose a new discrete dynamical system which provides a flexible model to fit population data. For different values of the three involved parameters, it can represent both globally persistent populations (compensatory or overcompensatory), and populations with Allee effects. In the most relevant cases of parameter values, there is a stable positive equilibrium, which is globally asymptotically stable in the persistent case. We study how population abundance depends on the parameters, and identify extinction windows between two saddle-node bifurcations.

In this paper, we study the dynamics of a non-autonomous semi-linear degenerate parabolic equation on \begin{document} $\mathbb{R}^N$ \end{document} driven by an unbounded additive noise. The nonlinearity has \begin{document} $(p,q)$ \end{document}-exponent growth and the degeneracy means that the diffusion coefficient $σ$ is unbounded and allowed to vanish at some points. Firstly we prove the existence of pullback attractor in \begin{document} $L^2(\mathbb{R}^N)$ \end{document} by using a compact embedding of the weighted Sobolev space. Secondly we establish the higher-attraction of the pullback attractor in \begin{document} $L^δ(\mathbb{R}^N)$ \end{document}, which implies that the cocycle is absorbing in \begin{document} $L^δ(\mathbb{R}^N)$ \end{document} after a translation by the complete orbit, for arbitrary \begin{document} $δ∈[2,∞)$ \end{document}. Thirdly we verify that the derived \begin{document} $L^2$ \end{document}-pullback attractor is in fact a compact attractor in \begin{document} $L^p(\mathbb{R}^N)\cap L^q(\mathbb{R}^N)\cap D_0^{1,2}(\mathbb{R}^N,σ)$ \end{document}, mainly by means of the estimate of difference of solutions instead of the usual truncation method.

The oscillation property of the Belousov-Zhabotinsky reaction and the color transition of its solution depend on the catalytic action of the metal ions. The solution of the reaction system catalyzed by both cerium ions and ferroin has a more complicated effect on the color than either the cerium-catalyzed case or the ferroin-catalyzed case. To theoretically elucidate the color transition of the case catalyzed by these two ions, a reduced model consisting of three differential equations is proposed, incorporating both the Rovinsky-Zhabotinsky scheme and the Field-Körös-Noyes scheme simplified by Tyson [Ann. N.Y. Acad. Sci., 316 (1979), pp.279-295]. The presented model can have a limit cycle under reasonable conditions through a Hopf bifurcation, and its existence theorem is proven by employing the bifurcation criterion established by Liu [J. Math. Anal. Appl., 182 (1994), pp.250-256].

In this paper, we study the modified Camassa-Holm (mCH) equation in Lagrangian coordinates. For some initial data $m_0$, we show that classical solutions to this equation blow up in finite time $T_{max}$. Before $T_{max}$, existence and uniqueness of classical solutions are established. Lifespan for classical solutions is obtained: $T_{max}≥ \frac{1}{||m_0||_{L^∞}||m_0||_{L^1}}.$ And there is a unique solution $X(ξ, t)$ to the Lagrange dynamics which is a strictly monotonic function of $ξ$ for any $t∈[0, T_{max})$: $X_ξ(·, t)>0$. As $t$ approaching $T_{max}$, we prove that the classical solution $m(·, t)$ in Eulerian coordinates has a unique limit $m(·, T_{max})$ in Radon measure space and there is a point $ξ_0$ such that $X_ξ(ξ_0, T_{max}) = 0$ which means $T_{max}$ is an onset time of collisions of characteristics. We also show that in some cases peakons are formed at $T_{max}$. After $T_{max}$, we regularize the Lagrange dynamics to prove global existence of weak solutions $m$ in Radon measure space.

We consider a free boundary problem modeling the growth of angiogenesis tumor with inhibitor, in which the tumor aggressiveness is modeled by a parameter \begin{document}$μ$\end{document}. The existences of radially symmetric stationary solution and symmetry-breaking stationary solution are established. In addition, it is proved that there exist a positive integer \begin{document}$m^{**}$\end{document} and a sequence of \begin{document}$μ_m$\end{document}, such that for each \begin{document}$μ_m(m > m^{**})$\end{document}, the symmetry-breaking stationary solution is a bifurcation branch of the radially symmetric stationary solution.

This paper concerns the existence of affine-periodic solutions for differential systems (including functional differential equations) and Newtonian systems with friction. This is a kind of pattern solutions in time-space, which may be periodic, anti-periodic, subharmonic or quasi periodic corresponding to rotation motions. Fink type conjecture is verified and Lyapunov's methods are given. These results are applied to study gradient systems and Newtonian (including Rayleigh or Lienard) systems. Levinson's conjecture to Newtonian systems is proved.

This work is devoted to study the spatial dynamics of a reaction-diffusion cholera model with spatial heterogeneity. In the case of the spatial domain is bounded and heterogeneous, we assume some key parameters in the model explicitly depend on spatial location. We first define the basic reproduction number \begin{document}$\mathcal{R}_0$\end{document} for the disease transmission, which generalizes the existing definition of \begin{document}$\mathcal{R}_0$\end{document} for the system in spatially homogeneous environment. Then we establish a threshold type result for the disease eradication (\begin{document}$\mathcal{R}_0 <1$\end{document}) or uniform persistence (\begin{document}$\mathcal{R}_0>1)$\end{document}. In the case of the domain is linear, unbounded, and spatially homogenerous, we further establish the existence of traveling wave solutions and the minimum wave speed \begin{document}$c^*$\end{document} for the disease transmission. At the end of this work, we characteristic the minimum wave speed \begin{document}$c^*$\end{document} and provide a method for the calculation of \begin{document}$c^*$\end{document}.