一类具耗散项的拟线性方程初值问题整体光滑解的存在性

THE EXISTENCE OF THE GLOBAL SMOOTH SOLUTION OF CAUCHY PROBLEM FOR A CLASS OF QUASILINER EQUTION WITH DISSIPATION

§1 序言本文考虑下述方程:这里 a>0是固定常数,σ:R→R,g:[0,+∞)×R→R,及 y0,y1:R→R 是给定的光滑函数,并假定:（σ）:σ∈C2（R）,σ（o）=0,σ′（ξ）≥ε>0 （ξ∈R;ε>0）且有σ″（ξ）≠0.（g）:g,gx∈C（[0,∞）×R),g（t）=（?）|g（t,x）|∈L∞（0,∞）∩ L′（0,∞）,

In this paper,we consider the following problem:(?)where α>0,σ″(ξ)≠0,σ′(ξ)≥ε>0,σ(0)=0.By giving suitable substitutionsand introducing the Riemann invarints,We can transform the above equationinto a strictly hyperbolic system of diagnoal form.By this method,we havemainly proved that Ifr_0(x)=y_1(x)+φ(y′_0(x))s_0(x)=y_1(x)-φ(y′_8(x))(where φ(v)=integral from 0 to v (σ(ξ))^(1/2)dξ)satisfy the monotonous condition:signσ″(v)·r_0(x)≤0 signσ″(v)·s_0(x)≤0then β~2-solution of cauchy problem (1)、(2) exists providedD=sup|y′_0(x)|+sup|y_1(x)|+‖g‖_(L′(0,∞))+‖g_1‖_(L~∞ (0,∞))is sufficiently small.