Well-posedness,global solutions and blowup phenomena for a nonlinearly dispersive wave equation

We establish the local well-posedness for a new nonlinearly dispersive wave equation which has solutions that exist for indefinite times as well as solutions that blowup infinite time. We also derive an explosion criterion for the equation, and we give a sharp estimate of the existence time for solutions with smooth initial data.     

Global existence and causality for a hyperbolic transmission problem with a repulsive nonlinearity

It is well known that the solution of the classical linear wave equation with an initial condition with compact support and vanishing initial velocity also has a compact support included in a set depending on time: the support of the solution at time t$t$ is causally related to that of the initial condition. Reed and Simon have shown that for a real-valued Klein–Gordon equation with (nonlinear) right-hand side −λu³$-\lambda u^3$ (λ>0$\lambda >0$), causality still holds. We show the same property for a one-dimensional Klein–Gordon problem but with transmission and with a more general repulsive nonlinear right-hand side F(u)$F\left(u\right)$. We also prove the global existence of a solution using the repulsiveness of F$F$. In the particular case F(u)=−λu³$F\left(u\right)=-\lambda u^3$, the problem is a relativistic model for a quantum particle with repulsive self-interaction and tunnel effect at a semi-infinite potential step.     

On the global existence of solutions to the Prandtl's system

In this paper we establish a global existence of weak solutions to the two-dimensional Prandtl's system for unsteady boundary layers in the class considered by Oleinik (J. Appl. Math. Mech. 30 (1966) 951) provided that the pressure is favourable. This generalizes the local well-posedness results due to Oleinik (1966; Mathematical Models in Boundary Layer Theory, Chapman & Hall, London, 1999). For the proof, we introduce a viscous splitting method so that the asymptotic behaviour of the solution near the boundary can be estimated more accurately by methods applicable to the degenerate parabolic equations.     

The blow-up curve of solutions of mixed problems for semilinear wave equations with exponential nonlinearities in one space dimension, I

In this paper, we consider mixed problems with a spacelike boundary derivative condition for semilinear wave equations with exponential nonlinearities in a quarter plane. Results similar to those obtained earlier by Caffarelli-Friedman for Cauchy problems and power nonlinearities are proved in the present situation, namely we show that solutions either are global or blow up on a spacelike curve. Weaker results are also obtained if the boundary vector field is tangent to the characteristic which leaves the domain in the future. Received January 7, 2000 / Accepted July 17, 2000 /Published online December 8, 2000     

Strongly damped wave equations with critical nonlinearities

This note deals with the strongly damped nonlinear wave equation

_utt−Δ_ut−Δu+f(_ut)+g(u)=h$u_{tt}-\Delta u_t-\Delta u+f\left(u_t\right)+g\left(u\right)=h$

with Dirichlet boundary conditions, where both the nonlinearities f$f$ and g$g$ exhibit a critical growth, while h$h$ is a time-independent forcing term. The existence of an exponential attractor of optimal regularity is proven. As a corollary, a regular global attractor of finite fractal dimension is obtained.     

Global energy conservative solutions to a system of variational wave equations

This paper is concerned with a system of variational wave equations which is the Euler–Lagrange equations of a variational principle arising in the theory of nematic liquid crystals and a few other physical contexts. The global existence of an energy-conservative weak solution to its Cauchy problem for initial data of finite energy is established by using the method of energy-dependent coordinates and the Young measure theory.     

Energy estimates for wave equations with time dependent propagation speeds in the Gevrey class

The total energy of the wave equation is conserved with respect to time if the propagation speed is a constant, but this is not true in general for time dependent propagation speeds. Indeed, it is considered in Hirosawa (2007) [3] that the following properties of the propagation speed are crucial for the estimates of the total energy: oscillating speed, difference from the mean, and the smoothness in Cm category. The main purpose of this paper is to derive a benefit of a further smoothness of the propagation speed in the Gevrey class for the energy estimates.     

On the asymptotic behavior of the energy for the wave equations with time depending coefficients

We consider the asymptotic behavior of the total energy of solutions to the Cauchy problem for wave equations with time dependent propagation speed. The main purpose of this paper is that the asymptotic behavior of the total energy is dominated by the following properties of the coefficient: order of the differentiability, behavior of the derivatives as t → ∞ and stabilization of the amplitude described by an integral. Moreover, the optimality of these properties are ensured by actual examples. Supported by Grants-in-Aid for Young Scientists (B) (No.16740098), The Ministry of Education, Culture, Sports, Science and Technology.     

含有快速增长非线性恢复力的梁方程的行波解

Abstract. On studying traveling waves on a nonlinearly suspended bridge,the following partial differential equation has been considered:     

Global existence and blowup of a localized problem with free boundary

This paper is concerned with a double fronts free boundary problem for the heat equation with a localized nonlinear reaction term. The local existence and uniqueness of the solution are given by applying the contraction mapping theorem. Then we present some conditions so that the solution blows up in finite time. Finally, the long-time behavior of the global solution is discussed. We show that the solution is global and fast if the initial data is small and that a global slow solution is possible when the initial data is suitably large.     

Global existence and uniform decay of a damped Klein-Gordon equation in a noncylindrical domain

In this paper, we consider a damped Klein-Gordon equation in a noncylindrical domain. This work is devoted to proving the existence of global solutions and decay for the energy of solutions for a damped Klein-Gordon equation in a noncylindrical domain.     

Results for a turbulent system with unbounded viscosities: Weak formulations,existence of solutions,boundedness and smoothness

We consider a circulation system arising in turbulence modelling in fluid dynamics with unbounded eddy viscosities. Various notions of weak solution are considered and compared. We establish existence and regularity results. In particular we study the boundedness of weak solutions. We also establish an existence result for a classical solution.     

Asymptotic profiles for wave equations with strong damping

We consider the Cauchy problem in Rⁿ${\mathbf{R}}^n$ for strongly damped wave equations. We derive asymptotic profiles of these solutions with weighted L^1,1(Rⁿ)$L^{1,1}({\mathbf{R}}^n)$ data by using a method introduced in [9] and/or [10].     

Global existence and blow-up phenomena for an integrable two-component Camassa-Holm shallow water system

This paper is concerned with global existence and blow-up phenomena for an integrable two-component Camassa-Holm shallow water system. A new global existence result and several new blow-up results of strong solutions to the system are presented. Our obtained results for the system are sharp and improve considerably earlier results.     

On existence, uniform decay rates and blow up for solutions of the 2-D wave equation with exponential source

This paper is concerned with the study of the nonlinear damped wave equation