The top-order energy of quasilinear wave equations in two space dimensions is uniformly bounded

Alinhac solved a long-standing open problem in 2001 and established that quasilinear wave equations in two space dimensions with quadratic null nonlinearities admit global-in-time solutions,provided that the initial data are compactly supported and sufficiently small in Sobolev norm.In this work,Alinhac obtained an upper bound with polynomial growth in time for the top-order energy of the solutions.A natural question then arises whether the time-growth is a true phenomenon,despite the possible conservation of basic energy.In the present paper,we establish that the top-order energy of the solutions in Alinhac theorem remains globally bounded in time.

INITIAL VALUE PROBLEMS AND FIRST BOUNDARY PROBLEMS FOR A CLASS OF QUASILINEAR WAVE EQUATIONS

The initial value problems and the first boundary problems for the quasilinear wave equation u_(tt)-[a_0+na_1(u_x)^(n-1)]u_(xx)-a_2u_(xxtt)=0 are considered,where a_0,a_2>0 are constants,a_1 is an arbitrary real number,n is a natural number.The existence and uniqueness of the classical solutions for the initial value problems and the first boundary problems of the equation (1) are proved by the Galerkin method.

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.

ON THE NUMERICAL SOLUTION OF QUASILINEAR WAVE EQUATION WITH STRONG DISSIPATIVE TERM

The numerical solution for a type of quasilinear wave equation is studied.The three-level difference scheme for quasi-linear waver equation with strong dissipative term is constructed and the convergence is proved.The error of the difference solution is estimated.The theoretical results are controlled on a numerical example.

GLOBAL EXISTENCE FOR THE NONHOMOGENEOUS QUASILINEAR WAVE EQUATION WITH A LOCALIZED WEAKLY NONLINEAR DISSIPATION IN EXTERIOR DOMAINS

It is proved that the global existence for the nonhomogeneous quasilinear wave equation with a localized weakly nonlinear dissipation in exterior domains.

Singularity Formation forthe General Poiseuille Flow of Nematic Liquid Crystals

We consider the Poiseuille flow of nematic liquid crystals via the full Ericksen-Leslie model.The model is described by a coupled system consisting of a heat equation and a quasilinear wave equation.In this paper,we will construct an example with a finite time cusp singularity due to the quasilinearity of the wave equation,extended from an earlier resultonaspecial case.

Stability of Planar Diffusion Wave for the Quasilinear Wave Equation with Nonlinear Damping

In this paper, we will show that under some smallness conditions, the planar diffusion wave v（x1/√＋t）is stable for a quasilinear wave equation with nonlinear damping：vtt-△f（v）＋vt＋g（vt）=0_3 x=（x1,x2…,xn）∈Rn,where v（x1/√1＋t）is the unique similar solution to the one dimensional nonlinear heat equa-tion：vt-f（v）x1 x1=0,f＇（v）〉0,v＋（∞,t）=v±,v＋≠v_.We also obtain the L∞ time decay rate whichreads‖v-v‖L∞=О（1）（1＋t）-r/4,where r=min（3,n）.To get the main result, the energy method and a newinequality have been used.

Decay Rate of Quasilinear Wave Equation with Viscosity

This paper is concerned with the decay rate of solutions for a quasilinear wave equation with viscosity. We use a so-called energy perturbation method to establish decay rate of solutions in terms of energy norm for a class of nonlinear functions. With the help of a comparison lemma of differential inequalities, we establish a relationship between decay rate of solutions and f.