Global existence and blow-up of solutions for a system of nonlinear viscoelastic wave equations with damping and source

In this paper we investigate the global existence and finite time blow-up of solutions to the system of nonlinear viscoelastic wave equations

in Ω×(0,T) with initial and Dirichlet boundary conditions, where Ω is a bounded domain in . Under suitable assumptions on the functions g_i(), , the initial data and the parameters in the equations, we establish several results concerning local existence, global existence, uniqueness and finite time blow-up property.     

The lifespan for nonlinear wave equations with multiple propagation speeds in 3D

In this paper, we consider the Cauchy problem for systems of nonlinear wave equations with multiple propagation speeds in three space dimensions. On the basis of some Klainerman–Sideris type weighted estimates, we get the lifespan of small amplitude solutions for systems with nonlinearities depending on both the unknown functions and their derivatives.     

Global existence of solutions for quasi-linear wave equations with viscous damping

In this paper, the global existence of solutions to the initial boundary value problem for a class of quasi-linear wave equations with viscous damping and source terms is studied by using a combination of Galerkin approximations, compactness, and monotonicity methods.     

Uniform bound of Sobolev norms of solutions to 3D nonlinear wave equations with null condition

This article concerns the time growth of Sobolev norms of classical solutions to the 3D quasi-linear wave equations with the null condition. Given initial data in H^s×H^s−1$H^s\times H^{s-1}$ with compact supports, the global well-posedness theory has been established independently by Klainerman [13] and Christodoulou [3], respectively, for a relatively large integer s  . However, the highest order Sobolev energy, namely, the H^s$H^s$ energy of solutions may have a logarithmic growth in time. In this paper, we show that the H^s$H^s$ energy of solutions is also uniformly bounded for s?5$s?5$. The proof employs the generalized energy method of Klainerman, enhanced by weighted L²$L^2$ estimates and the ghost weight introduced by Alinhac.     

Global existence of small solutions for quadratic quasilinear Klein-Gordon systems in two space dimensions

Consider a quasi-linear system of two Klein-Gordon equations with masses m₁, m₂. We prove that when m₁≠2m₂ and m₂≠2m₁, such a system has global solutions for small, smooth, compactly supported Cauchy data. This extends a result proved by Sunagawa (J. Differential Equations 192 (2) (2003) 308) in the semi-linear case. Moreover, we show that global existence holds true also when m₁=2m₂ and a convenient null condition is satisfied by the nonlinearities.     

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.     

Nonresonance and Global Existence of Nonlinear Elastic Wave Equations with External Forces

This paper establishes the global existence of classical solution to the system of homogeneous,isotropic hyperelasticity with time-independent external force,provided that the nonlinear term obeys a ty...     

Global existence and nonexistence of solution for Cauchy problem of two-dimensional generalized Boussinesq equations

In this paper we consider the Cauchy problem of two-dimensional generalized Boussinesq-type equation _utt−Δu−Δ_utt+Δ²u+Δf(u)=0$u_{tt}-\mathrm{\Delta }u-\mathrm{\Delta }{u}_{tt}+{\mathrm{\Delta }}^{2}u+\mathrm{\Delta }f(u)=0$. Under the assumption that f(u)$f(u)$ is a function with exponential growth at infinity and under some assumptions on the initial data, we prove the existence and nonexistence of global weak solution. There are very few works on Boussinesq equation with nonlinear exponential growth term by potential well theory.     

Global existence and nonexistence for some degenerate and quasilinear parabolic systems

The author discusses the degenerate and quasilinear parabolic system

     

Global existence and blowup solutions for quasilinear parabolic equations

The authors discuss the quasilinear parabolic equation ut=∇⋅(g(u)∇u)+h(u,∇u)+f(u) with u_|∂Ω=0, u(x,0)=?(x). If f, g and h are polynomials with proper degrees and proper coefficients, they show that the blowup property only depends on the first eigenvalue of −Δ in Ω with Dirichlet boundary condition. For a special case, they obtain a sharp result.     

Existence and nonexistence of a global solution for coupled nonlinear wave equations with damping and source

We consider the system of nonlinear wave equations

     

Global existence, asymptotic behavior and blowup of solutions for a class of nonlinear wave equations with dissipative term

The paper studies the global existence, asymptotic behavior and blowup of solutions to the initial boundary value problem for a class of nonlinear wave equations with dissipative term. It proves that under rather mild conditions on nonlinear terms and initial data the above-mentioned problem admits a global weak solution and the solution decays exponentially to zero as t→+∞, respectively, in the states of large initial data and small initial energy. In particular, in the case of space dimension N=1, the weak solution is regularized to be a unique generalized solution. And if the conditions guaranteeing the global existence of weak solutions are not valid, then under the opposite conditions, the solutions of above-mentioned problem blow up in finite time. And an example is given.     

Global existence of null-form wave equations on small asymptotically Euclidean manifolds

We prove the global existence of the small solutions to the Cauchy problem for quasilinear wave equations satisfying the null condition on (R³,g)$({\mathbb{R}}^3,\mathfrak{g})$, where the metric g$\mathfrak{g}$ is a small perturbation of the flat metric and approaches the Euclidean metric like (1+|x|²)^−ρ/2${(1+{|x|}^2)}^{-\rho /2}$ with ρ>1$\rho >1$. Global and almost global existence for systems without the null condition are also discussed for certain small time-dependent perturbations of the flat metric in Appendix?A.     

Global existence and nonexistence of solutions for a system of nonlinear damped wave equations

We consider the Cauchy problem for the system of semilinear damped wave equations with small initial data:

We show that a critical exponent which classifies the global existence and the finite time blow up of solutions indeed coincides with the one to a corresponding semilinear heat systems with small data. The proof of the global existence is based on the L^p–L^q estimates of fundamental solutions for linear damped wave equations [K. Nishihara, L^p–L^q estimates of solutions to the damped wave equation in 3-dimensional space and their application, Math. Z. 244 (2003) 631–649; K. Marcati, P. Nishihara, The L^p–L^q estimates of solutions to one-dimensional damped wave equations and their application to compressible flow through porous media, J. Differential Equations 191 (2003) 445–469; T. Hosono, T. Ogawa, Large time behavior and L^p–L^q estimate of 2-dimensional nonlinear damped wave equations, J. Differential Equations 203 (2004) 82–118; T. Narazaki, L^p–L^q estimates for damped wave equations and their applications to semilinear problem, J. Math. Soc. Japan 56 (2004) 585–626]. And the blow-up is shown by the Fujita–Kaplan–Zhang method [Q. Zhang, A blow-up result for a nonlinear wave equation with damping: The critical case, C. R. Acad. Sci. Paris 333 (2001) 109–114; F. Sun, M. Wang, Existence and nonexistence of global solutions for a nonlinear hyperbolic system with damping, Nonlinear Anal. 66 (12) (2007) 2889–2910; T. Ogawa, H. Takeda, Non-existence of weak solutions to nonlinear damped wave equations in exterior domains, Nonlinear Anal. 70 (10) (2009) 3696–3701].     

Global existence to generalized Boussinesq equation with combined power-type nonlinearities

The Cauchy problem to the generalized Boussinesq equation with combined power-type nonlinearities is studied. Global solvability or finite time blow-up of the solutions with subcritical initial energy is proved by means of the sign preserving property of the Nehari functional. For generalized Lienard (or generalized Bernoulli) nonlinear terms the critical energy constant is explicitly evaluated. A new method, that can be considered as a modification of the potential well method, is developed. The performed numerical experiments support the theoretical results.     

Global nonexistence for logarithmic wave equations with nonlinear damping and distributed delay terms

This paper is concerned with global nonexistence of solutions for a logarithmic wave equation with nonlinear damping and distributed delay terms. Due to the simultaneous presence of nonlinear damping and logarithmic source terms, we have difficulty in use of the concavity method. Applying the energy estimates, we show the global nonexistence of solutions with not only non-positive initial energy but also positive initial energy.     

Global existence for nonlinear Klein-Gordon equations in infinite homogeneous waveguides in two dimensions

In this paper we prove a global existence result for nonlinear Klein-Gordon equations with small data in infinite homogeneous waveguids, R²×M, where M=(M,g) is a Zoll manifold. The method is based on the normal forms, the eigenfunction expansion for M and the special distribution of eigenvalues of Laplace-Beltrami on Zoll manifold.     

非线性Klein-Gordon方程整体解存在的最佳条件

本文研究一类带竞争势函数的非线性Klein-Gordon方程的柯西问题.根据基 态的特征,运用势井方法和凹方法导出了该问题解爆破和整体存在的最佳条件. 同时 还回答了当初值为多小时,整体解存在这个问题.