Global existence and asymptotic behavior of solution for Rosenau equation with Stokes damped term

We study the small‐data Cauchy problem for n‐dimensional Stokes damped Rosenau equation. Under some assumptions, we prove the global existence and uniqueness of the small‐amplitude solution by utilizing the contraction mapping principle and study the asymptotic behavior of the solution. Copyright © 2015 John Wiley & Sons, Ltd.     

Large‐time behavior of solutions to a 1D liquid crystal system

In this paper, we prove the large‐time behavior, as time tends to infinity, of solutions in $H^i\times H_0^i\times H^{i+1}(i=1,2)$ and $H^4\times H_0^4\times H^4$ for a system modeling the nematic liquid crystal flow, which consists of a subsystem of the compressible Navier‐Stokes equations coupling with a subsystem including a heat flow equation for harmonic maps.     

Asymptotic profile of solutions for the damped wave equation with a nonlinear convection term

This paper is concerned with the large time behavior of solutions to the initial value problem for the damped wave equations with nonlinear convection in one‐dimensional whole space. In 2007, Ueda and Kawashima showed that the solution tends to a self similar solution of the Burgers equation. However, they did not mention that their decay estimate is optimal or not. Under this situation, the aim of this paper was to find out the sharp decay estimate by studying the second asymptotic profile of solutions. The explicit representation formula and the decay estimates of the solution for the linearized equation including the lower order term play crucial roles in our analysis.     

Large time behavior of solutions for a Cauchy problem on a nonlinear conservation law with large initial data in the whole space

We consider the Cauchy problem on a nonlinear conversation law with large initial data. By Green's function methods, energy methods, Fourier analysis, and frequency decomposition, we obtain the global existence and the optimal time‐decay estimate of solutions.     

Boundary value problem for a global‐in‐time parabolic equation

The aim of this paper is to draw attention to an interesting semilinear parabolic equation that arose when describing the chaotic dynamics of a polymer molecule in a liquid. This equation is nonlocal in time and contains a term, called the interaction potential, that depends on the time‐integral of the solution over the entire interval of solving the problem. In fact, one needs to know the “future” in order to determine the coefficient in this term, that is, the causality principle is violated. The existence of a weak solution of the initial boundary value problem is proven. The interaction potential satisfies fairly general conditions and can have arbitrary growth at infinity. The uniqueness of this solution is established with restrictions on the length of the considered time interval.     

Blowup and decay behavior of solutions to the generalized Boussinesq‐type equation with strong damping

In this paper, we study the Cauchy problem for the generalized Boussinesq‐type equation with strong damping. By defining a suitable solution space with time‐weighted norms and under smallness condition on the initial data, we establish the global existence and decay property of the solutions. Under certain conditions on the initial data, we also provide blowup of the solutions.     

Global existence of solutions for damped wave equations with nonlinear memory

This article studies the Cauchy problem for the damped wave equation with nonlinear memory. For a noncompactly supported initial data with small energy, global existence and asymptotic behaviour of solutions are obtained when 1?≤?n?≤?3. This result generalized the previous result by Fino [Critical exponent for damped wave equations with nonlinear memory, Nonlinear Anal. 74 (2011), pp. 5495–5505], which dealt with the solution with compactly supported initial data.     

A space‐time mixed‐hybrid finite element method for the damped wave equation

A space‐time finite element method is introduced to solve the linear damped wave equation. The scheme is constructed in the framework of the mixed‐hybrid finite element methods, and where an original conforming approximation of H(div;Ω) is used, the latter permits us to obtain an upwind scheme in time. We establish the link between the nonstandard finite difference scheme recently introduced by Mickens and Jordan and the scheme proposed. In this regard, two approaches are considered and in particular we employ a formulation allowing the solution to be marched in time, i.e., one only needs to consider one time increment at a time. Numerical results are presented and compared with the analytical solution illustrating good performance of the present method. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008     

Large‐time behaviour of solutions to non‐linear wave equations: higher‐order asymptotics

The Cauchy problems for the Korteweg–de Vries–Burgers equation and the Benjamin–Bona– Mahony–Burgers equation are studied. Using subtle estimates of solutions to the linearized equations, the higher‐order terms of the asymptotic expansion as of solutions are derived. Copyright © 1999 John Wiley & Sons, Ltd.     

The large time behavior of solutions of a diffusion equation involving a nonlocal convection term

We study the existence, uniqueness, and asymtotic behaviour of non-negative solutions to a parabolic diffusion equation involving a nonlocal con-vective term on a bounded domain ? contained in Rⁿ. We apply the In variance Principle of LaSalle and Hale to prove that the solution tends to zero as t → ∞     

Asymptotic behavior of solutions to space‐time fractional diffusion‐reaction equations

This article discusses the analyticity and the long‐time asymptotic behavior of solutions to space‐time fractional diffusion‐reaction equations in . By a Laplace transform argument, we prove that the decay rate of the solution as t→∞ is dominated by the order of the time‐fractional derivative. We consider the decay rate also in a bounded domain. Copyright © 2016 John Wiley & Sons, Ltd.     

Large time behavior of solutions to a class of doubly nonlinear parabolic equations

We study the large time asymptotic behavior of solutions of the doubly degenerate parabolic equation u _t = div(u ^m−1|Du| ^p−2 Du) − u ^q with an initial condition u(x, 0) = u ₀(x). Here the exponents m, p and q satisfy m + p ⩾ 3, p > 1 and q > m + p − 2. The paper was supported by NSF of China (10571144), NSF for youth of Fujian province in China (2005J037) and NSF of Jimei University in China.     

Large time behavior of solutions for a system of nonlinear damped wave equations

We consider the Cauchy problem of the semilinear damped wave system:

     

Gauss‐Lobatto‐Legendre‐Birkhoff pseudospectral approximations for the multi‐term time fractional diffusion‐wave equation with Neumann boundaryconditions

A second‐order finite difference/pseudospectral scheme is proposed for numerical approximation of multi‐term time fractional diffusion‐wave equation with Neumann boundary conditions. The scheme is based upon the weighted and shifted Grünwald difference operators approximation of the time fractional calculus and Gauss‐Lobatto‐Legendre‐Birkhoff (GLLB) pseudospectral method for spatial discretization. The unconditionally stability and convergence of the scheme are rigorously proved. Numerical examples are carried out to verify theoretical results.     

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.     

Large time behavior of solutions to a nonlinear hyperbolic relaxation system with slowly decaying data

We consider the large time asymptotic behavior of the global solutions to the initial value problem for the nonlinear damped wave equation with slowly decaying initial data. When the initial data decay fast enough, it is known that the solution to this problem converges to the self-similar solution to the Burgers equation called a nonlinear diffusion wave, and its optimal asymptotic rate is obtained. In this paper, we focus on the case that the initial data decay more slowly than previous works and derive the corresponding asymptotic profile. Moreover, we investigate how the change of the decay rate of the initial values affect its asymptotic rate.     

General decay of solutions to a viscoelastic wave equation with linear damping,nonlinear damping and source term

ABSTRACT

This paper is concerned with the decay property of a nonlinear viscoelastic wave equation with linear damping, nonlinear damping and source term. Under weaker assumption on the relaxation function, we establish a general decay result, which extends the result obtained in Messaoudi [Exponential decay of solutions of a nonlinearly damped wave equation. Nodea-Nonlinear Differ Equat Appl. 2005;12:391–399].     

Construction of solutions of the defocusing nonlinear Schrödinger equation with asymptotically time‐periodic boundary values

We study the defocusing nonlinear Schrödinger equation in the quarter plane with asymptotically periodic boundary values. We use the unified transform method, also known as the Fokas method, and the Deift‐Zhou nonlinear steepest descent method to construct solutions in a sector close to the boundary whose leading behavior is described by a single exponential plane wave. Furthermore, we compute the subleading terms in the long‐time asymptotics of the constructed solutions.     

The initial value problem for a multi‐dimensional radiation hydrodynamics model with viscosity

In this paper, we study the existence and time‐asymptotic behavior of solutions to the Cauchy problem for the equations of radiation hydrodynamics with viscosity in ?³. The global existence of the solutions is obtained by using the energy method. With more elaborate energy estimates, we also give some decay rates of the solutions. Copyright © 2010 John Wiley & Sons, Ltd.