ASYMPTOTIC STABILITY OF SOLUTION TO THE INITIAL-BOUNDARY VALUE PROBLEM FOR SCALAR VISCOUS CONSERVATION LAWS CORRESPONDING TO RAREFACTION WAVES

1.IntroductionWeconsidertheinitial-boundaryvalueproblem(IBVP)forscalarviscousconservationlaw(ut f(u).=uxx,xE(0, co)=R ,t>0,o,(n \--9,ifI~9,f~M(IBVP){u(0't)=u~(t)-1~i---t:7),u(x,0)=u000={=:0)f:LO .,wheref"(u)>0foralluunderconsideration,andu--u (1.1),andu~(t)~u~EL'(R ),U'(t)EL'(R ).(1.2)Undercondition(1.l)or(l.l)',thecorrespondingRiemannproblem{;;ot:<:>;i,>=x{,:3'f:,:(13)yieldstherarefaCtionwavesolution(u~)x5f'(u~)t,r"(X,t)=17j,)~1(…

ASYMPTOTIC STABILITY OF SOLUTION TO THE INITIAL-BOUNDARY VALUE PROBLEM FOR SCALAR VISCOUS CONSERVATION LAWS CORRESPONDING TO RAREFACTION WAVES

This paper is concerned with the asymptotic behaviors of solutions of thegeneral initial-boundary value problem to scalar viscous conservation law uxx on R with the conditions u(0, t) = u-(t) u-(t ),u(x,0) = u_0(x) u (x-), where f"(u) > 0 for all u under consideration, u- < u , and u--(t) u(R ), Here the corresponding Riemann problem 0, u0(x) = u- when x <0 and u0(x)=u when x > 0 admits the rarefaction wave. Ourproblem is divided into five cases depending on the signs of the characteristic speeds of the boundary state u- at t= and the far field state u = u( ). Both the globalexistence of the solution and the asymptotic stability are shown in all cases.