The asymptotic behaviour of solutions for a class of quasilinear wave equations with cubic nonlinearity in two space dimensions

For a class of quasilinear wave equations with small initial data, first we give the lower bound of lifespan of classical solutions, then we discuss the long time asymptotic behaviour of solutions away from the blowup time. This project is supported by the Tianyuan Foundation of China and Laburay of Mathematics for Nonlinear Problems, Fudan University.     

Blow Up of Solutions to Semilinear Wave Equations with Critical Exponent in High Dimensions

In this paper, the author considers the Cauchy problem for semilinear wave equations with critical exponent in n≥4 space dimensions. Under some positivity conditions on the initial data, it is proved that there can be no global solutions no matter how small the initial data are.     

BLOWUP OF SOLUTIONS TO THE CAUCHY PROBLEM FOR NONLINEARWAVE EQUATIONS

gi. IntroductionThis paper deals with solutionS of certain nonlinear wave equationS Of the formcorresponding to Antial conditionSwuersis the wave OPerstor.we are interested in showing the ~ up" Of solutions to (1.1)--(1.2). For that, wereIf ac ~ 1)(n ~ 1) > 2, global solutions of ~ equation subject to very general perturbationsof order p exist Provided the initial data are swhciently small (see I6] and references therein).We are also interested in esthaattw the take when "blow up" occurs. …     

BLOWUP OF SOLUTIONS TO THE CAUCHY PROBLEM FOR NONLINEARWAVE EQUATIONS

The author proves blow up of solutions to the Cauchy problem of certain nonlinear wave equations and, also, estimates the time when the blow up occurs.     

The blowup for 2-D quasilinear wave equations with cubic nonlinearity

For 2-D quasilinear wave equations with cubic nonlinearity and small initial data, we not only show that the solutions blow up in finite time but also give a complete asymptotic expansion of the lifespan of classical solutions. Hence we solve a problem posed by S. Alinhac and A. Hoshiga. Moreover, as an application of this result, we prove the blowup of solutions for the nonlinear vibrating membrane equations.     

非线性波动方程解的Blow up

本文主要讨论高维空间非线性波动方程的Ｃａｕｃｈｙ问题整体解的非存在性，我们证明对Ｕｕ－△ｕ＝ｆ（ｕ），ｆ（ｕ）＝ｃ｜ｕ｜ｐ－１ｕ，当１＜ｐ≥时，若初始能量非正，则无论初值数据的ｃｋ－范数（连续空间范数）多么小，解按Ｃｋ－范数或按Ｈｓ（Ｒｎ）（Ｓ≥１）都在有限时间内Ｂｌｏｗｕｐ，并且有相同的生命跨度．     

Exact boundary controllability for quasilinear wave equations in a planar tree-like network of strings

In this paper the local exact boundary controllability for quasilinear wave equations on a planar tree-like network of strings is established and the number of boundary controls is equal to the number of simple nodes minus 1.     

Cauchy problem for quasi-linear wave equations with viscous damping

The paper studies the global existence and asymptotic behavior of weak solutions to the Cauchy problem for quasi-linear wave equations with viscous damping. It proves that when pmax{m,α}, where m+1, α+1 and p+1 are, respectively, the growth orders of the nonlinear strain terms, the nonlinear damping term and the source term, the Cauchy problem admits a global weak solution, which decays to zero according to the rate of polynomial as t→∞, as long as the initial data are taken in a certain potential well and the initial energy satisfies a bounded condition. Especially in the case of space dimension N=1, the solutions are regularized and so generalized and classical solution both prove to be unique. Comparison of the results with previous ones shows that there exist clear boundaries similar to thresholds among the growth orders of the nonlinear terms, the states of the initial energy and the existence, asymptotic behavior and nonexistence of global solutions of the Cauchy problem.     

Nonlinear wave equations and singular solutions

We construct solutions to nonlinear wave equations that are singular along a prescribed noncharacteristic hypersurface, which is the graph of a function satisfying not the Eikonal but another partial differential equation of the first order. The method of Fuchsian reduction is employed.

     

Sharpness on the Lower Bound of the Lifespan of Solutions to Nonlinear Wave Equations

This paper is devoted to proving the sharpness on the lower bound of the lifespan of classical solutions to general nonlinear wave equations with small initial data in the case n = 2 and cubic nonlinearity (see the results of T. T. Li and Y. M. Chen in 1992). For this purpose, the authors consider the following Cauchy problem:

Higher approximations of the averaging method for quasilinear parabolic equations with fast oscillating principal part in the case of the Cauchy problem

An algorithm for constructing the higher approximations of the averaging method for quasilinear parabolic equations with fast oscillating coefficients is suggested and justified. Translated fromMatematicheskie Zametki, Vol. 65, No. 4, pp. 562–572, April, 1999.     

Life span of positive solutions for a semilinear heat equation with general non-decaying initial data

We prove upper bounds on the life span of positive solutions for a semilinear heat equation. For non-decaying initial data, it is well known that the solutions blow up in finite time. We give two types estimates of the life span in terms of the limiting values of the initial data in space.     

Cauchy problem for quasi-linear wave equations with nonlinear damping and source terms

The paper studies the existence and non-existence of global weak solutions to the Cauchy problem for a class of quasi-linear wave equations with nonlinear damping and source terms. It proves that when α?max{m,p}, where m+1, α+1 and p+1 are, respectively, the growth orders of the nonlinear strain terms, the nonlinear damping term and the source term, under rather mild conditions on initial data, the Cauchy problem admits a global weak solution. Especially in the case of space dimension N=1, the weak solutions are regularized and so generalized and classical solution both prove to be unique. On the other hand, if 0?α<1, and the initial energy is negative, then under certain opposite conditions, any weak solution of the Cauchy problem blows up in finite time. And an example is shown.     

Global existence of classical solutions to the Cauchy problem for a kind of quasilinear hyperbolic systems in divergence form

This paper deals with the global existence of classical solutions to a kind of second order quasilinear hyperbolic systems subject to a null condition, with the linear elastodynamic system as its principal part and the nonlinear terms depending on the product of u² and the derivatives of u.     

Exact boundary controllability of nodal profile for 1-D quasilinear wave equations

Based on the theory of semi-global C ² solution for 1-D quasilinear wave equations, the local exact boundary controllability of nodal profile for 1-D quasilinear wave equations is obtained by a constructive method, and the corresponding global exact boundary controllability of nodal profile is also obtained under certain additional hypotheses.