Search

This paper is concerned with two classes of cubic quasilinear equations, which can be derived as asymptotic models from shallow-water approximation to the 2D incompressible Euler equations. One class of the models has homogeneous cubic nonlinearity and includes the integrable modified Camassa–Holm (mCH) equation and Novikov equation, and the other class encompasses both quadratic and cubic nonlinearities. It is demonstrated here that both these models possess localized peaked solutions. By constructing a Lyapunov function, these peaked waves are shown to be dynamically stable under small perturbations in the natural energy space $H^1$, without restriction on the sign of the momentum density. In particular, for the homogeneous cubic nonlinear model, we are able to further incorporate a higher-order conservation law to conclude orbital stability in $H^1\cap W^{1,4}$. Our analysis is based on a strong use of the conservation laws, the introduction of certain auxiliary functions, and a refined continuity argument.

This paper is devoted to the initial and boundary value problems for a class of nonlinear metaparabolic equations u t − β u x x − k u x x t + γ u x x x x = f u x x . At low initial energy level ( J u 0 < d ), we not only prove the existence of global weak solutions for these problems by the combination of the Galerkin approximation and potential well methods but also obtain the finite time blow-up result by adopting the potential well and improved concavity skills. Finally, we also discussed the finite time blow-up phenomenon for certain solutions of these problems with high initial energy.

In this paper, we investigate the orbital stability of solitary waves for the following generalized long-short wave resonance equations of Hamiltonian form: $$ \textstyle\begin{cases} iu_{t}+u_{{xx}}=\alpha uv+\gamma \vert u \vert ^{2}u+\delta \vert u \vert ^{4}u, \\ v_{t}+\beta \vert u \vert ^{2}_{x}=0. \end{cases} $$ { i u t + u x x = α u v + γ | u | 2 u + δ | u | 4 u , v t + β | u | x 2 = 0 . We first obtain explicit exact solitary waves for Eqs. (0.1). Second, by applying the extended version of the classical orbital stability theory presented by Grillakis et al., the approach proposed by Bona et al., and spectral analysis, we obtain general results to judge orbital stability of solitary waves. We finally discuss the explicit expression of $\det (d^{\prime \prime })$ det ( d ″ ) in three cases and provide specific orbital stability results for solitary waves. Especially, we can get the results obtained by Guo and Chen with parameters $\alpha =1$ α = 1 , $\beta =-1$ β = − 1 , and $\delta =0$ δ = 0 . Moreover, we can obtain the orbital stability of solitary waves for the classical long-short wave equation with $\gamma =\delta =0$ γ = δ = 0 and the orbital instability results for the nonlinear Schrödinger equation with $\beta =0$ β = 0 .

This paper investigates an elliptic entropy of uncertain random variables and its application in the area of portfolio selection. We first define the elliptic entropy to characterize the uncertainty of uncertain random variables and give some mathematical properties of the elliptic entropy. Then we derive a computational formula to calculate the elliptic entropy of function of uncertain random variables. Furthermore, we use the elliptic entropy to characterize the risk of investment and establish a mean-entropy portfolio selection model, in which the future security returns are described by uncertain random variables. Based on the chance theory, the equivalent form of the mean–entropy model is derived. To show the performance of the mean–entropy portfolio selection model, several numerical experiments are presented. We also numerically compare the mean–entropy model with the mean–variance model, the equi-weighted portfolio model, and the most diversified portfolio model by using three kinds of diversification indices. Numerical results show that the mean-entropy model outperforms the mean–variance model in selecting diversified portfolios no matter of using which diversification index.

This paper deals with a class of nonlocal semilinear pseudo-parabolic equation with conical degeneration u t − △ B u t − △ B u = | u | p − 1 u − 1 | B | ∫ B | u | p − 1 u d x 1 x 1 d x ′ , on a manifold with conical singularity, where △ B is Fuchsian type Laplace operator with totally characteristic degeneracy on the boundary x 1 = 0 . By using the modified method of potential well with Galerkin approximation and concavity, the global existence, uniqueness, finite time blow up and asymptotic behavior of the solutions will be discussed at the low initial energy J ( u 0 ) d and critical initial energy J ( u 0 ) = d , respectively. Furthermore, we investigate the global existence and finite time blow up of the solutions with the high initial energy J ( u 0 ) > d by the variational method. Especially, we also derive the threshold results of global existence and nonexistence for the solutions at two different initial energy levels, i.e. low initial level and critical initial level.

This paper deals with the blow-up phenomena for a type of nonlinear porous medium equations with weighted source ut −4um = a(x)f(u) subject to Dirichlet (or Neumann) boundary conditions. Based on the auxiliary functions and differential-integral inequalities, the blow-up criterions which ensure that u cannot exist all time are given under two different assumptions, and the corresponding estimates on the upper bounds for blow-up time and blow-up rate are derived respectively. Moreover, we use three different methods to determine the lower bounds for blow-up time and blow-up rate estimates if blow-up does occurs.

This paper is devoted to study the long-time dynamics for a nonlinear viscoelastic Kirchhoff plate equation. Under some growth conditions of g and f, the existence of a global attractor is granted. Furthermore, in the subcritical case, this global attractor has finite Hausdorff and fractal dimensions.

In this paper, we consider the initial boundary value problem for the fourth order wave equation with nonlinear boundary velocity feedbacks \begin{document}$ f_{1}(u_{\nu{t}}) $\end{document} , \begin{document}$ f_{2}(u_{t}) $\end{document} and internal source \begin{document}$ |u|^{\rho}u $\end{document} . Under some geometrical conditions, the existence and uniform decay rates of the solutions are proved even if the nonlinear boundary velocity feedbacks \begin{document}$ f_{1}(u_{\nu{t}}) $\end{document} , \begin{document}$ f_{2}(u_{t}) $\end{document} have not polynomial growth near the origin respectively. By the combination of the Galerkin approximation, potential well method and a special basis constructed, we first obtain the global existence and uniqueness of regular solutions and weak solutions. In addition, we also investigate the explicit decay rate estimates of the energy, the ideas of which are based on the construction of a special weight function \begin{document}$ \phi(t) $\end{document} (that depends on the behaviors of the functions \begin{document}$ f_{1}(u_{\nu{t}}) $\end{document} , \begin{document}$ f_{2}(u_{t}) $\end{document} near the origin), nonlinear integral inequality and the Multiplier method.

This paper deals with the blow-up phenomena for a nonlinear pseudo-parabolic equation with a memory term $u_{t}-\triangle{u}-\triangle{u}_{t}+\int_{0}^{t}g(t-\tau)\triangle{u}(\tau)d\tau = |{u}|^{p}{u}$ in a bounded domain, with the initial and Dirichlet boundary conditions. We first obtain the finite time blow-up results for the solutions with initial data at non-positive energy level as well as arbitrary positive energy level, and give some upper bounds for the blow-up time $T^{*}$ depending on the sign and size of initial energy $E(0)$. In addition, a lower bound for the life span $T^{*}$ is derived by means of a differential inequality technique if blow-up does occur.

This paper deals with a class of nonlinear viscoelastic wave equation with damping and source terms ${u_{tt}} - \Delta u - \Delta {u_t} - \Delta {u_{tt}} + \int_0^t {g(t - s)\Delta u(s)ds + {u_t}{u_t}{^{m - 2}} = u|u|{^{p - 2}}} $ with acoustic boundary conditions. Under some appropriate assumption on relaxation function g and the initial data, we prove that the solution blows up in finite time if the positive initial energy satisfies a suitable condition.

This paper deals with the initial boundary value problem for the nonlinear beam equation with double damping terms $$\begin{aligned} u_{tt}-u_{xxt}+u_{xxxx}+u_{xxxxt}=g(u_{xx})_{xx}, \quad x\in\Omega, t>0, \end{aligned}$$ where $\Omega=(0,1)$ , and $g(s)$ is a given nonlinear function. We derive sufficient conditions for the blow-up of the solution to the problem by virtue of an adapted concavity method. In addition, global existence of weak solutions as well as exponential and uniform decay rates of the solution energy are established by the use of an integral inequality.

This paper concerns with the Cauchy problem for two classes of nonlinear hyperbolic equations with double damping terms. Firstly, by virtue of the Fourier transform method, we prove that the Cauchy problem of a class of high order nonlinear hyperbolic equation admits a global smooth solution $u(x, t)\in C^{\infty}((0, T]; H^{\infty}(\mathbb{R}))$$\bigcap C([0, T]; H^{3}(\mathbb{R}))$$\bigcap C^{1}([0, T]; H^{-1}(\mathbb{R}))$ as long as initial value $u_{0}\in W^{4, 1}(\mathbb{R})\bigcap H^{3}(\mathbb{R}), u_{1}\in L^{1}(\mathbb{R})\bigcap H^{-1}(\mathbb{R})$. Moreover, we give the sufficient conditions on the blow-up of the solution of a nonlinear damped hyperbolic equation with the initial value conditions in finite time and an example.

We classify the existence and non-existence of a class of localized solitary waves for the cubic Camassa–Holm-type equation which arises as an asymptotic model with a nonlocal cubic nonlinearity for the unidirectional propagation of shallow water waves. In addition to those peaked solitary-wave solutions, we show by the phase portrait method that there are smooth and cusped solitary waves according to the wave speed and coefficients of nonlocal linear and nonlinear dispersions. We then prove that the smooth solitary-wave solutions are orbitally stable in the energy space under small perturbation. Finally, we establish a Liouville-type property of the solution.

We consider a pseudo-parabolic equation with nonlinearities of variable exponent type u t − ν △ u t − div ( | ∇ u | m ( x ) − 2 ∇ u ) = | u | p ( x ) − 2 u , in Ω × ( 0 , T ) , associated with initial and Dirichlet boundary conditions. By means of a differential inequality technique, we obtain an upper bound for blow-up time if variable exponents p ( ⋅ ) , m ( ⋅ ) and the initial data satisfy some conditions. Also, a lower bound for blow-up time is determined under some other conditions.

This paper deals with the initial boundary value problem for a metaparabolic equations withtime dependent coeffcient. Under suitable conditions on initial data, a blow-up criterion which ensuresthat u cannot exist all time is given, and an upper bound for blow up time is derived. Moreover, wealso obtain a lower bound for blow-up time if blow up does occur by means of a di erential inequalitytechnique.

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.

In this paper, we consider the initial boundary value problem for a viscoelastic wave equation with nonlinear boundary source term. First of all, we introduce a family of potential wells and prove the invariance of some sets. Then we establish the existence and nonexistence of global weak solution with small initial energy under suitable assumptions on the relaxation function , nonlinear function , the initial data and the parameters in the equation. Furthermore, we obtain the global existence of weak solution for the problem with critical initial conditions and .

In this paper, we consider the initial boundary value problem for a fourth order pseudo-parabolic equation with memory and source terms. Under suitable assumptions on the function $g$, the initial data and the parameters in the equation, we not only prove the existence of global weak solutions by the combination of the Galerkin method and potential well theory, but also establish an explicit decay rate estimate of the energy adopting the ideas of Marcelo M. Cavalcanti et al. (J. Differ. Equations 203 (2004) 119-158) and Patrick Martinez (ESAIN: Control Optim. Calc. Var. 4 (1999) 419-444). Furthermore, the finite time blow up results for the solutions with both positive and negative initial energy are obtained under certain conditions.

This paper deals with the initial boundary value problem for strongly damped semilinear wave equations with logarithmic nonlinearity u t t − Δ u − Δ u t = φ p ( u ) log | u | in a bounded domain Ω ⊂ R n . We discuss the existence, uniqueness and polynomial or exponential energy decay estimates of global weak solutions under some appropriate conditions. Moreover, we derive the finite time blow up results of weak solutions, and give the lower and upper bounds for blow-up time by the combination of the concavity method, perturbation energy method and differential–integral inequality technique.

We consider the existence and nonexistence of global weak solutions to the Cauchy problem for a higher order generalized Boussinesq-type equation with hydrodynamical damped term in -dimensional space. The existence of global weak solutions is proved under the assumptions that the nonlinear term is polynomial growth order, either , , the constant , or the initial data belongs to the potential well. Moreover, we establish two finite-time blow up results for any weak solutions with negative initial energy or nonnegative initial energy using the concavity method.

This paper investigates the existence and uniqueness of time-periodic solutions of the periodic initial value problem for the modified Zakharov equations with a quantum correction. By combining a priori estimates with the Galerkin method and Leray–Schauder fixed point theorem, we prove that there exist a unique strong time-periodic solution and a unique classical time-periodic solution under some conditions on the forcing terms f and g.

We consider the nonlinear pseudoparabolic equation with a memory termut-Δu-Δut+∫0tλt-τΔuτdτ=div∇up-2u+u1+α,x∈Ω,t>0, with an initial condition and Dirichlet boundary condition. Under negative initial energy and suitable conditions onp,α, and the relaxation functionλ(t), we prove a finite-time blow-up result by using the concavity method.

In this paper, we consider the initial boundary value problem of the double dispersive-dissipative wave equation with nonlinear damping and source terms. By the combination of the Galerkin method and the monotonicity-compactness method, the existence of global solutions is obtained with the least amount of a priori estimates. Moreover, the asymptotic behavior of global solutions is investigated under some assumptions on the initial data.

We consider the blow-up phenomenon of sixth order nonlinear strongly damped wave equation. By using the concavity method, we prove a finite time blow-up result under assumptions on the nonlinear term and the initial data.