Nonlinear waves in friedman-robertson-walker space-times

In this paper we consider the initial value problem for the nonlinear wave equation □u = F(u, u′) in Friedman-Robertson-Walker space-time, □ being the D'Alambertian in local coordinates of space-time. We obtain decay estimates and show that the equation has global solutions for small initial data. We do it by reducing the problem to an initial value problem for the wave equation over hyperbolic space. As byproduct we derive decay and global existence for solutions of the wave equation over the hyperbolic space with small initial data. The same technique with some auxiliary lemmas similar to the ones proved in [6], [7] can be used to generalize the result to the case when F depends also on second derivatives of u in a certain way.     

On the global existence and small dispersion limit for a class of complex Ginzburg–Landau equations

In this paper we consider a class of complex Ginzburg–Landau equations. We obtain sufficient conditions for the existence and uniqueness of global solutions for the initial‐value problem in d‐dimensional torus ??^d, and that solutions are initially approximated by solutions of the corresponding small dispersion limit equation for a period of time that goes to infinity as dispersive coefficient goes to zero. Copyright © 2008 John Wiley & Sons, Ltd.     

Global existence and asymptotic behavior of solutions for a semi-linear wave equation

In this paper, we investigate the initial value problem for a semi-linear wave equation in n-dimensional space. Based on the decay estimate of solutions to the corresponding linear equation, we define a set of time-weighted Sobolev spaces. Under small condition on the initial value, we prove the global existence and asymptotic behavior of the solution in the corresponding Sobolev spaces by the contraction mapping principle.     

On the asymptotic behavior of solution for the generalized IBq equation with Stokes damped term

In this paper, we investigate the Cauchy problem for the generalized improved Boussinesq equation with Stokes damped term in n-dimensional space. We observe that the dissipative structure of the linearized equation is of the regularity-loss type. This means that we have the optimal decay estimates of solutions under the additional regularity assumption on the initial data. Based on the decay estimates of solutions to the corresponding linear equation and smallness condition on the initial data, we prove the global existence and asymptotic of the small amplitude solution in the time-weighted Sobolev space by the contraction mapping principle.     

Point‐wise decay estimate for the global classical solutions to quasilinear hyperbolic systems

In this paper, we first consider the Cauchy problem for quasilinear strictly hyperbolic systems with weak linear degeneracy. The existence of global classical solutions for small and decay initial data was established in (Commun. Partial Differential Equations 1994; 19 :1263–1317; Nonlinear Anal. 1997; 28 :1299–1322; Chin. Ann. Math. 2004; 25B :37–56). We give a new, very simple proof of this result and also give a sharp point‐wise decay estimate of the solution. Then, we consider the mixed initial‐boundary‐value problem for quasilinear hyperbolic systems with nonlinear boundary conditions in the first quadrant. Under the assumption that the positive eigenvalues are weakly linearly degenerate, the global existence of classical solution with small and decay initial and boundary data was established in (Discrete Continuous Dynamical Systems 2005; 12 (1):59–78; Zhou and Yang, in press). We also give a simple proof of this result as well as a sharp point‐wise decay estimate of the solution. Copyright © 2008 John Wiley & Sons, Ltd.     

Large time behavior of solutions to the nonlinear pseudo-parabolic equation

In this paper, we investigate the initial value problem for the nonlinear pseudo-parabolic equation. Global existence and optimal decay estimate of solution are established, provided that the initial value is suitably small. Moreover, when n?2$n?2$ and the nonlinear term f(u)$f(u)$ disappears, we prove that the global solutions can be approximated by the linear solution as time tends to infinity. When n=1$n=1$ and the nonlinear term f(u)$f(u)$ disappears, we show that as time tends to infinity, the global solution approaches the nonlinear diffusion wave described by the self-similar solution of the viscous Burgers equation.     

An existence theorem for the unmodified Vlasov Equation

The initial value problem for the modified Vlasov equation with a mollification parameter δ > 0, as introduced by Batt, has a unique global solution in the weak sense whenever f₀ ε L¹ and f₀ ≧ 0 λ-a.e. Assuming boundedness of f₀ and boundedness of the kinetic energy, it is shown that, as δ → 0, there are subsequences δ_n → 0 such that the corresponding solutions converge weakly in the measure-theoretical sense. The limits are shown to be global weak solutions of the initial value problem for Vlasov's equation, and these solutions are seen to be weakly continuous with respect to t. For the plasma physical case, boundedness of the kinetic energy is a consequence of energy conservation.     

On the asymptotic behavior of solution for the generalized IBq equation with hydrodynamical damped term

We study the Cauchy problem for the generalized IBq equation with hydrodynamical damped term in n-dimensional space. We observe that the dissipative structure of the linearized equation is of the regularity-loss type. This means that we have the optimal decay estimates of solutions under the additional regularity assumption on the initial data. Under smallness condition on the initial data, we prove the global existence and decay of the small amplitude solution in the Sobolev space.     

The Cauchy–Neumann problem for a multi‐dimensional isentropic hydrodynamic model for semiconductors

We investigate a multi‐dimensional isentropic hydrodynamic (Euler–Poisson) model for semiconductors, where the energy equation is replaced by the pressure–density relation p(n) . We establish the global existence of smooth solutions for the Cauchy–Neumann problem with small perturbed initial data and homogeneous Neumann boundary conditions. We show that, as t→+∞, the solutions converge to the non‐constant stationary solutions of the corresponding drift–diffusion equations. Moreover, we also investigate the existence and uniqueness of the stationary solutions for the corresponding drift–diffusion equations. Copyright © 2005 John Wiley & Sons, Ltd.     

Asymptotic behavior of solutions to a class of nonlinear wave equations of sixth order with damping

We investigate the initial value problem for a class of nonlinear wave equations of sixth order with damping. The decay structure of this equation is of the regularity‐loss type, which causes difficulty in high‐frequency region. By using the Fourier splitting frequency technique and energy method in Fourier space, we establish asymptotic profiles of solutions to the linear equation that is given by the convolution of the fundamental solutions of heat and free wave equation. Moreover, the asymptotic profile of solutions shows the decay estimate of solutions to the corresponding linear equation obtained in this paper that is optimal under some conditions. Finally, global existence and optimal decay estimate of solutions to this equation are also established. Copyright © 2016 John Wiley & Sons, Ltd.     

Some limit analysis of a three dimensional viscous compressible capillary model for plasma

In this study, we discuss some limit analysis of a viscous capillary model of plasma, which is expressed as a so‐called the compressible Navier‐Stokes‐Poisson‐Korteweg equation. First, the existence of global smooth solutions for the initial value problem to the compressible Navier‐Stokes‐Poisson‐Korteweg equation with a given Debye length λ and a given capillary coefficient κ is obtained. We also show the uniform estimates of global smooth solutions with respect to the Debye length λ and the capillary coefficient κ. Then, from Aubin lemma, we show that the unique smooth solution of the 3‐dimensional Navier‐Stokes‐Poisson‐Korteweg equations converges globally in time to the strong solution of the corresponding limit equations, as λ tends to zero, κ tends to zero, and λ and κ simultaneously tend to zero. Moreover, we also give the convergence rates of these limits for any given positive time one by one.     

On the Global Solution of a 3‐D MHD System with Initial Data near Equilibrium

In this paper, we prove the global existence of smooth solutions to the three‐dimensional incompressible magnetohydrodynamical system with initial data close enough to the equilibrium state, (e₃,0). Compared with previous works by Lin, Xu, and Zhang and by Xu and Zhang, here we present a new Lagrangian formulation of the system, which is a damped wave equation and which is nondegenerate only in the direction of the initial magnetic field. Furthermore, we remove the admissible condition on the initial magnetic field, which was required in the earlier works. By using the Frobenius theorem and anisotropic Littlewood‐Paley theory for the Lagrangian formulation of the system, we achieve the global L¹‐in‐time Lipschitz estimate of the velocity field, which allows us to conclude the global existence of solutions to this system. In the case when the initial magnetic field is a constant vector, the large‐time decay rate of the solution is also obtained.© 2016 Wiley Periodicals, Inc.     

Global existence to a reaction–diffusion equation through a self‐similar‐like upper solution

A demonstration method is presented, which will ensure the existence of positive global solutions in time to the reaction–diffusion equation ?u_t+Δu+u^p=0 in ?ⁿ×[0, ∞), for exponents p?3 and space dimensions n?3. This method does not require the initial value to have a specific uniform smallness condition, but rather to satisfy a bell‐like form. The method is based on a specific upper solution, which models the diffusion process of the heat equation. The upper solution is not self‐similar, but does have a self‐similar‐like form. After transforming the reaction–diffusion problem into an equivalent one, whose initial value is uniformly very small, a local solution is obtained in the time interval [0, 1] by the use of this upper solution. This local solution is then extended to [0, ∞) through an infinite sequence of extensions. At each step, an appropriate change of variables will transform the extension into a problem nearly identical to the local problem in [0, 1]. These transformations exploit the diffusive and self‐similar‐like nature of the upper solution. Copyright © 2009 John Wiley & Sons, Ltd.     

Nonlinear Fourier Transforms and the mKdV Equation in the Quarter Plane

The unified transform method introduced by Fokas can be used to analyze initial‐boundary value problems for integrable evolution equations. The method involves several steps, including the definition of spectral functions via nonlinear Fourier transforms and the formulation of a Riemann‐Hilbert problem. We provide a rigorous implementation of these steps in the case of the mKdV equation in the quarter plane under limited regularity and decay assumptions. We give detailed estimates for the relevant nonlinear Fourier transforms. Using the theory of L²‐RH problems, we consider the construction of quarter plane solutions which are C¹ in time and C³ in space.     

On the global weak solutions for a modified two‐component Camassa‐Holm equation

In this paper, we investigate the existence of global weak solutions to the Cauchy problem of a modified two‐component Camassa‐Holm equation with the initial data satisfying lim_{x → ±∞}u₀(x) = u_±. By perturbing the Cauchy problem around a rarefaction wave, we obtain a global weak solution for the system under the assumption u_? ≤ u₊. The global weak solution is obtained as a limit of approximation solutions. The key elements in our analysis are the Helly theorem and the estimation of energy for approximation solutions in $H^1(\mathbb {R})\times H^1(\mathbb {R})In this paper, we investigate the existence of global weak solutions to the Cauchy problem of a modified two‐component Camassa‐Holm equation with the initial data satisfying lim_{x → ±∞}u₀(x) = u_±. By perturbing the Cauchy problem around a rarefaction wave, we obtain a global weak solution for the system under the assumption u_? ≤ u₊. The global weak solution is obtained as a limit of approximation solutions. The key elements in our analysis are the Helly theorem and the estimation of energy for approximation solutions in $H^1(\mathbb {R})\times H^1(\mathbb {R})$ and some a priori estimates on the first‐order derivatives of approximation solutions.     

On the asymptotic behaviour of solution for the generalized double dispersion equation

In this article, we investigate the Cauchy problem for the generalized double dispersion equation in n-dimensional space. We establish the decay estimates of solution to the corresponding linear equation. Under smallness condition on the initial data, we prove the global existence and asymptotic behaviour of the small amplitude solution in the time-weighted Sobolev space by the contraction mapping principle.     

The threshold of effective damping for semilinear wave equations

In this paper, we obtain the global existence of small data solutions to the Cauchy problem in space dimension n ≥ 1, for p > 1 + 2 ∕ n, where μ is sufficiently large. We obtain estimates for the solution and its energy with the same decay rate of the linear problem. In particular, for μ ≥ 2 + n, the damping term is effective with respect to the L¹ ? L² low‐frequency estimates for the solution and its energy. In this case, we may prove global existence in any space dimension n ≥ 3, by assuming smallness of the initial data in some weighted energy space. In space dimension n = 1,2, we only assume smallness of the data in some Sobolev spaces, and we introduce an approach based on fractional Sobolev embedding to improve the threshold for global existence to μ ≥ 5 ∕ 3 in space dimension n = 1 and to μ ≥ 3 in space dimension n = 2. Copyright © 2014 John Wiley & Sons, Ltd.     

Global nonexistence for an integro‐differential equation

The initial boundary value problem for an integro‐differential equation with nonlinear damping and source terms in a bounded domain is considered. By modifying the method in a work by Autuori et al. in 2010, we establish the nonexistence result of global solutions with the initial energy controlled by a critical value. This improves earlier results in the literatures. Copyright © 2011 John Wiley & Sons, Ltd.     

Existence and asymptotic behaviour of solutions for a class of quasi‐linear evolution equations with non‐linear damping and source terms

We consider a class of quasi‐linear evolution equations with non‐linear damping and source terms arising from the models of non‐linear viscoelasticity. By a Galerkin approximation scheme combined with the potential well method we prove that when m<p, where m(?0) and p are, respectively, the growth orders of the non‐linear strain terms and the source term, under appropriate conditions, the initial boundary value problem of the above‐mentioned equations admits global weak solutions and the solutions decay to zero as t→∞. Copyright © 2002 John Wiley & Sons, Ltd.     

The asymptotics of blow-up solutions to a degenerate parabolic equation

Summary In this paper a quasilinear parabolic equation with strong nonlinearity is studied. This equation may have solutions which blow up in finite time, a global behaviour which has been studied extensively in recent years. By using the invariant property of equation and the scaling invariant solutions-self-similar solutions it is proved that the solutions of the Cauchy problem inR ⁿ with a class of specified initial values will blow up in finite time at a single point.