Introduction

In this note we consider the existence of global solution to the initial value problem for the quasilinear wave equation:

$$u_{tt}-( 1+\Vert\nabla u\Vert^2)\Delta u +\rho(u_t)=0 \quad\hbox{in}\quad R^N \times (0,\infty), \eqno{(1.1)}$$

with

$$u(x,0)= u_0(x) \quad\hbox{and}\quad u_t(x,0)= u_1(x), \eqno{(1.2)}$$

where $\Vert\cdot\Vert$ denotes $L^2$ norm in $R^N$ and $\rho(v)$ is a differentiable function satisfying

$$0 < k_0 \leq \rho'(v) \leq k_1 < \infty \quad\hbox{and}\quad \rho(0)=0. \eqno{(1.3)}$$

For the Kirchhoff type quasilinear wave equation it is natural to seek for the solutions in the class $C([0,\infty); H_2) \cap C^1([0,\infty);H_1) \cap C^2([0,\infty); L^2)$ or a little weaker space $L^\infty([0,\infty);$ $H_2) \cap W^{1,\infty}([0,\infty);H_1)\cap W^{2,\infty}([0,\infty);L^2)$ (cf. I. Lasiecka and J.Ong [2]), and we are interested in the global solution of the problem (1.1)-(1.2) in such a class, ( we often call such a solution as $H^2$ solution). When $\rho(u_t)=u_t$, linear, we see $\rho(u_t)u={1\over2} \frac{d}{dt} \vert u\vert^2$ and by use of this fact we can easily derive the a priori estimates

$$\Vert u_t(t)\Vert^2+\Vert\nabla u(t)\Vert^2 \leq C(\Vert u_0\Vert _{H_1}+\Vert u_1\Vert _{L^2})(1+t)^{-1}$$

and

$$\Vert u_{tt}(t)\Vert^2+\Vert\nabla u_t(t)\Vert^2 \leq C(\Vert u_0\Vert _{H_2}+\Vert u_1\Vert )(1+t)^{-2}$$

if $\Vert u_0\Vert _{H_2}+\Vert u_1\Vert _{H_1}$ is small. These a priori estimates are sufficient for the desired global solution. Indeed, K. Mochizuki [5] has proved such result under a more general condition on the dissipation. However, the proof heavily depends on the linearity of the dissipation $\rho(u_t)$ and cannot be applied to the case of nonlinear dissipation. Y. Yamada [9] proved the existence of global solutions without direct use of the decay properties. But in [9] also, the linearity of the dissipation is essentially used. The object of this note is to prove the existence of global $H^2$-solution when $\rho(v)$ is weakly nonlinear as (1.3), though in our case, solution $u(t)$ itself belongs to $L_{loc}^\infty([0,\infty);L^2)$, not $L^\infty([0,\infty);L^2)$. Our proof is based on the following observations. First, we see for an assumed $H^2$-solution $u(t)$,

$$E(t)+ \int_0^t \int_{ R^N} \rho(u_t(s)) u_t(s) dxds =E(0),$$

where

$$E(t)={1\over2}[\Vert u_t(t)\Vert^2+(1+\Vert\nabla u(t)\Vert^2)\Vert\nabla u(t)\Vert^2]$$

and hence

$$k_0 \int_0^\infty \Vert u_t(s)\Vert^2 ds \leq E(0) < \infty. \eqno{(1.4)}$$

Next, differentiating the equation we have

$$u_{ttt}-( 1+\Vert\nabla u\Vert^2)\Delta u_t-2(\nabla u, \nabla u_t) \Delta u +\rho'(u_t)u_{tt}=0,$$

which is rewritten as

$$U_{tt}-( 1+\Vert\nabla u\Vert^2)\Delta U+\rho'(u_t)U_{t}=2(\nabla u, \nabla u_t) \Delta u$$

with $U=u_t$. Since $\rho'(u_t) U_t$ is linear in $U_t$, we can expect the decay estimate

$$\Vert U_{t}(t)\Vert^2+\Vert\nabla U(t)\Vert^2 \leq C(\Vert u_0\Vert _{H_2}+\Vert u_1\Vert)(1+t)^{-1} \eqno{(1.5)}$$

if $(u_0, u_1)$ is small. This estimate is weaker than the case of linear dissipation, but combining this with (1.4) we have a hope to get desired $H_2$-solutions. We notice that if the problem is considered in a bounded domain $\Omega$ with the boundary condition $u\vert _{\partial \Omega}=0$, it is easy to derive exponential decay

$$E(t) \leq C(E(0)) e^{-\lambda t}, \quad \lambda > 0,$$

and the global existence of $H_2$- solution is easily proved. In fact, more general problem have been treated by many authors (Lasiecka and Ong [2], T. Mizumachi[4], Ono [7,8], Matsuyama and Ikehata [3], Nishihara and Yamada [6] etc.). But the problem in the whole space $R^N$ is more delicate because of the lack of Poincaré inequality. When $\rho(u_t) \equiv 0$, there are proved deep results on global existence and scattering by Yamazaki [10], where Fourier integral method is employed.