American Institute of Mathematical Sciences

The paper studies the asymptotic behaviour of solutions to a second-order non-linear discrete equation of Emden–Fowler type

\begin{document}$ \Delta^2 u(k) \pm k^\alpha u^m(k) = 0 $\end{document}

where \begin{document}$ u\colon \{k_0, k_0+1, \dots\}\to \mathbb{R} $\end{document} is an unknown solution, \begin{document}$ \Delta^2 u(k) $\end{document} is its second-order forward difference, \begin{document}$ k_0 $\end{document} is a fixed integer and \begin{document}$ \alpha $\end{document}, \begin{document}$ m $\end{document} are real numbers, \begin{document}$ m\not = 0, 1 $\end{document}.

In this paper, we consider a class of finitely degenerate coupled parabolic systems. At high initial energy level \begin{document}$ J(u_{0})>d $\end{document}, we present a new sufficient condition to describe the global existence and nonexistence of solutions for problem (1)-(4) respectively. Moreover, by applying the Levine's concavity method, we give some affirmative answers to finite time blow up of solutions at arbitrary positive initial energy \begin{document}$ J(u_{0})>0 $\end{document}, including the estimate of upper bound of blowup time.

The Cauchy problem of one dimensional generalized Boussinesq equation is treated by the approach of variational method in order to realize the control aim, which is the control problem reflecting the relationship between initial data and global dynamics of solution. For a class of more general nonlinearities we classify the initial data for the global solution and finite time blowup solution. The results generalize some existing conclusions related this problem.

We consider a class of nonlinear evolution equations of second order in time, linearly damped and with a memory term. Particular cases are viscoelastic wave, Kirchhoff and Petrovsky equations. They appear in the description of the motion of deformable bodies with viscoelastic material behavior. Several articles have studied the nonexistence of global solutions of these equations due to blow-up. Most of them have considered non-positive and small positive values of the initial energy and recently some authors have analyzed these equations for any positive value of the initial energy. Within an abstract functional framework we analyze this problem and we improve the results in the literature. To this end, a new positive invariance set is introduced.

In this paper, we study axisymmetric homogeneous solutions of the Navier-Stokes equations in cone regions. In [James Serrin. The swirling vortex. Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, 271(1214):325-360, 1972.], Serrin studied the boundary value problem in half-space minus \begin{document}$ x_3 $\end{document}-axis, and used it to model the dynamics of tornado. We extend Serrin's work to general cone regions minus \begin{document}$ x_3 $\end{document}-axis. All axisymmetric homogeneous solutions of the boundary value problem have three possible patterns, which can be classified by two parameters. Some existence results are obtained as well.

In this paper, we study a reaction-diffusion SEI epidemic model with/without immigration of infected hosts. Our results show that if there is no immigration for the infected (exposed) individuals, the model admits a threshold behaviour in terms of the basic reproduction number, and if the system includes the immigration, the disease always persists. In each case, we explore the global attractivity of the equilibrium via Lyapunov functions in the case of spatially homogeneous environment, and investigate the asymptotic behavior of the endemic equilibrium (when it exists) with respect to the small migration rate of the susceptible, exposed or infected population in the case of spatially heterogeneous environment. Our results suggest that the strategy of controlling the migration rate of population can not eradicate the disease, and the disease transmission risk will be underestimated if the immigration of infected hosts is ignored.

Considered herein is the well-posedness and stability for the Cauchy problem of the fourth-order Schrödinger equation with nonlinear derivative term \begin{document}$ iu_{t}+\Delta^2 u-u\Delta|u|^2+\lambda|u|^pu = 0 $\end{document}, where \begin{document}$ t\in\mathbb{R} $\end{document} and \begin{document}$ x\in \mathbb{R}^n $\end{document}. First of all, for initial data \begin{document}$ \varphi(x)\in H^2(\mathbb{R}^{n}) $\end{document}, we establish the local well-poseness and finite time blow-up criterion of the solutions, and give a rough estimate of blow-up time and blow-up rate. Secondly, under a smallness assumption on the initial value \begin{document}$ \varphi(x) $\end{document}, we demonstrate the global well-posedness of the solutions by applying two different methods, and at the same time give the scattering behavior of the solutions. Finally, based on founded a priori estimates, we investigate the stability of solutions by the short-time and long-time perturbation theories, respectively.

This paper deals with the sixth-order Boussinesq equation with fourth-order dispersion term and nonlinear source. By using some ordinary differential inequalities, the conditions on finite time blow-up of solutions are given with suitable assumptions on initial values. Moreover, the upper and lower bounds of the blow-up time are also investigated.

In this paper, we study the fractional pseudo-parabolic equations \begin{document}$ u_{t} + \left(-\Delta\right)^{s} u + \left(-\Delta\right)^{s} u_{t} = u\log \left| u \right| $\end{document}. Firstly, we recall the relationship between the fractional Laplace operator \begin{document}$ \left(-\Delta\right)^{s} $\end{document} and the fractional Sobolev space \begin{document}$ H^{s} $\end{document} and discuss the invariant sets and the vacuum isolating behavior of solutions with the help of a family of potential wells. Then, we derive a threshold result of existence of weak solution: for the low initial energy case (i.e., \begin{document}$ J(u_{0}) < d $\end{document}), the solution is global in time with \begin{document}$ I(u_{0}) >0 $\end{document} or \begin{document}$ \Vert u_{0}\Vert_{{X_{0}(\Omega)}} = 0 $\end{document} and blows up at \begin{document}$ +\infty $\end{document} with \begin{document}$ I(u_{0}) < 0 $\end{document}; for the critical initial energy case (i.e., \begin{document}$ J(u_{0}) = d $\end{document}), the solution is global in time with \begin{document}$ I(u_{0}) \geq0 $\end{document} and blows up at \begin{document}$ +\infty $\end{document} with \begin{document}$ I(u_{0}) < 0 $\end{document}. The decay estimate of the energy functional for the global solution is also given.

In this paper, the initial-boundary value problem for a class of fourth-order nonlinear parabolic equations modeling the epitaxial growth of thin films is studied. By means of the theory of potential wells, the global existence, asymptotic behavior and finite time blow-up of weak solutions are obtained.

In this paper, for the damped generalized incompressible Navier-Stokes equations on \begin{document}$ \mathbb{T}^{2} $\end{document} as the index \begin{document}$ \alpha $\end{document} of the general dissipative operator \begin{document}$ (-\Delta)^{\alpha} $\end{document} belongs to \begin{document}$ (0,\frac{1}{2}] $\end{document}, we prove the absence of anomalous dissipation of the long time averages of entropy. We also give a note to show that, by using the \begin{document}$ L^{\infty} $\end{document} bounds given in Caffarelli et al. [4], the absence of anomalous dissipation of the long time averages of energy for the forced SQG equations established in Constantin et al. [12] still holds under a slightly weaker conditions \begin{document}$ \theta_{0}\in L^{1}(\mathbb{R}^{2})\cap L^{2}(\mathbb{R}^{2}) $\end{document} and \begin{document}$ f \in L^{1}(\mathbb{R}^{2})\cap L^{p}(\mathbb{R}^{2}) $\end{document} with some \begin{document}$ p>2 $\end{document}.

This paper is devoted to studying the dynamical stability of periodic peaked solitary waves for the generalized modified Camassa-Holm equation. The equation is a generalization of the modified Camassa-Holm equation and it possesses the Hamiltonian structure shared by the modified Camassa-Holm equation. The equation admits the periodic peakons. It is shown that the periodic peakons are dynamically stable under small perturbations in the energy space.

We consider the well-posedness of solution of the initial boundary value problem to the fourth order wave equation with the strong and weak damping terms, and the logarithmic strain term, which was introduced to describe many complex physical processes. The local solution is obtained with the help of the Galerkin method and the contraction mapping principle. The global solution and the blowup solution in infinite time under sub-critical initial energy are also established, and then these results are extended in parallel to the critical initial energy. Finally, the infinite time blowup of solution is proved at the arbitrary positive initial energy.

We consider an anisotropic double phase problem with a reaction in which we have the competing effects of a parametric singular term and a superlinear perturbation. We prove a bifurcation-type result describing the changes in the set of positive solutions as the parameter varies on \begin{document}$ \mathring{\mathbb{R}}_+ = (0, +\infty) $\end{document}. Our approach uses variational tools together with truncation and comparison techniques as well as several general results of independent interest about anisotropic equations, which are proved in the Appendix.

In this paper, we are concerned with a reaction-diffusion SIS epidemic model with saturated incidence rate, linear source and spontaneous infection mechanism. We derive the uniform bounds of parabolic system and obtain the global asymptotic stability of the constant steady state in a homogeneous environment. Moreover, the existence of the positive steady state is established. We mainly analyze the effects of diffusion, saturation and spontaneous infection on the asymptotic profiles of the steady state. These results show that the linear source and spontaneous infection can enhance the persistence of an infectious disease. Our mathematical approach is based on topological degree theory, singular perturbation technique, the comparison principles for elliptic equations and various elliptic estimates.

The paper deals with global existence and blow-up results for a class of fourth-order wave equations with nonlinear damping term and superlinear source term with the coefficient depends on space and time variable. In the case the weak solution is global, we give information on the decay rate of the solution. In the case the weak solution blows up in finite time, estimate the lower bound and upper bound of the lifespan of the blow-up solution, and also estimate the blow-up rate. Finally, if our problem contains an external vertical load term, a sufficient condition is also established to obtain the global existence and general decay rate of weak solutions.

In this paper, we study fractional subdiffusion fourth parabolic equations containing Caputo and Caputo-Fabrizio operators. The main results of the paper are presented in two parts. For the first part with the Caputo derivative, we focus on the global and local well-posedness results. We study the global mild solution for biharmonic heat equation with Caputo derivative in the case of globally Lipschitz source term. A new weighted space is used for this case. We then proceed to give the results about the local existence in the case of locally Lipschitz source term. To overcome the intricacies of the proofs, we applied \begin{document}$ L^p-L^q $\end{document} estimate for biharmonic heat semigroup, Banach fixed point theory, some estimates for Mittag-Lefler functions and Wright functions, and also Sobolev embeddings. For the second result involving the Cahn-Hilliard equation with the Caputo-Fabrizio operator, we first show the local existence result. In addition, we first provide that the connections of the mild solution between the Cahn-Hilliard equation in the case \begin{document}$ 0<{\alpha}<1 $\end{document} and \begin{document}$ {\alpha} = 1 $\end{document}. This is the first result of investigating the Cahn-Hilliard equation with this type of derivative. The main key of the proof is based on complex evaluations involving exponential functions, and some embeddings between \begin{document}$ L^p $\end{document} spaces and Hilbert scales spaces.

The aim of this paper is to give global nonexistence and blow–up results for the problem

\begin{document}$ \begin{cases} u_{tt}-\Delta u+P(x,u_t) = f(x,u) \qquad &\text{in $(0,\infty)\times\Omega$,}\\ u = 0 &\text{on $(0,\infty)\times \Gamma_0$,}\\ u_{tt}+\partial_\nu u-\Delta_\Gamma u+Q(x,u_t) = g(x,u)\qquad &\text{on $(0,\infty)\times \Gamma_1$,}\\ u(0,x) = u_0(x),\quad u_t(0,x) = u_1(x) & \text{in $\overline{\Omega}$,} \end{cases} $\end{document}

where \begin{document}$ \Omega $\end{document} is a bounded open \begin{document}$ C^1 $\end{document} subset of \begin{document}$ {\mathbb R}^N $\end{document}, \begin{document}$ N\ge 2 $\end{document}, \begin{document}$ \Gamma = \partial\Omega $\end{document}, \begin{document}$ (\Gamma_0,\Gamma_1) $\end{document} is a partition of \begin{document}$ \Gamma $\end{document}, \begin{document}$ \Gamma_1\not = \emptyset $\end{document} being relatively open in \begin{document}$ \Gamma $\end{document}, \begin{document}$ \Delta_\Gamma $\end{document} denotes the Laplace–Beltrami operator on \begin{document}$ \Gamma $\end{document}, \begin{document}$ \nu $\end{document} is the outward normal to \begin{document}$ \Omega $\end{document}, and the terms \begin{document}$ P $\end{document} and \begin{document}$ Q $\end{document} represent nonlinear damping terms, while \begin{document}$ f $\end{document} and \begin{document}$ g $\end{document} are nonlinear source terms. These results complement the analysis of the problem given by the author in two recent papers, dealing with local and global existence, uniqueness and well–posedness.

In this paper, we deal with the initial boundary value problem of the following fractional wave equation of Kirchhoff type

\begin{document}$ \begin{align*} u_{tt}+M([u]_{\alpha, 2}^2)(-\Delta)^{\alpha}u+(-\Delta)^{s}u_{t} = \int_{0}^{t}g(t-\tau)(-\Delta)^{\alpha}u(\tau)d\tau+\lambda|u|^{q -2}u, \end{align*} $\end{document}

where \begin{document}$ M:[0, \infty)\rightarrow (0, \infty) $\end{document} is a nondecreasing and continuous function, \begin{document}$ [u]_{\alpha, 2} $\end{document} is the Gagliardo-seminorm of \begin{document}$ u $\end{document}, \begin{document}$ (-\Delta)^\alpha $\end{document} and \begin{document}$ (-\Delta)^s $\end{document} are the fractional Laplace operators, \begin{document}$ g:\mathbb{R}^+\rightarrow \mathbb{R}^+ $\end{document} is a positive nonincreasing function and \begin{document}$ \lambda $\end{document} is a parameter. First, the local and global existence of solutions are obtained by using the Galerkin method. Then the global nonexistence of solutions is discussed via blow-up analysis. Our results generalize and improve the existing results in the literature.

This paper studies the Cauchy problem of Schrödinger equation with inhomogeneous nonlinear term \begin{document}$ V(x)|\varphi|^{p-1}\varphi $\end{document} in \begin{document}$ \mathbb{R}^n $\end{document}. For the case \begin{document}$ p>1+\frac{4(1+\varepsilon_0)}{n} (0<\varepsilon_0<\frac{2}{n-2}) $\end{document}, by introducing a potential well, we obtain some invariant sets of solution and give a sharp condition of global existence and finite time blowup of solution; for the case \begin{document}$ p<1+\frac{4}{n} $\end{document}, we obtain the global existence of solution for any initial data in \begin{document}$ H^1 (\mathbb{R}^n) $\end{document}.

We mainly focus on the asymptotic behavior analysis for certain fourth-order nonlinear wave equations with strain term, nonlinear weak damping term and source term. We establish two theorems on the asymptotic behavior of the solution depending on some conditions related to the relationship among the forced strain term, the nonlinear weak damping term and source terms.

This paper considers the Cauchy problem for a 2-component Camassa-Holm system

\begin{document}$ \begin{equation*} m_t = ( u m)_x+ u _xm- v m, \ \ n_t = ( u n)_x+ u _xn+ v n, \end{equation*} $\end{document}

where \begin{document}$ n+m = \frac{1}{2}( u _{xx}-4 u ) $\end{document}, \begin{document}$ n-m = v _x $\end{document}, this model was proposed in [2] from a novel method to the Euler-Bernoulli Beam on the basis of an inhomogeneous matrix string problem. The local well-posedness in Sobolev spaces \begin{document}$ H^s(\mathbb{R})\times H^{s-1}(\mathbb{R}) $\end{document} with \begin{document}$ s>\frac{5}{2} $\end{document} of this system was investigated through the Kato's theory, then the blow-up criterion for this system was described by the technique on energy methods. Finally, we established the analyticity in both time and space variables of the solutions for this system with a given analytic initial data.