The authors consider the asymptotic behavior of solutions of the quasilinear hyperbolic equation with linear damping u_(tt)+u_(t)-div(a(u)u)=0,and show that,at least when n≥3,they tend,as t-+∞,to those of the nonlin...The authors consider the asymptotic behavior of solutions of the quasilinear hyperbolic equation with linear damping u_(tt)+u_(t)-div(a(u)u)=0,and show that,at least when n≥3,they tend,as t-+∞,to those of the nonlinear parabolic equation v_t-div(a(v)v)=0,in the sense that the norm||u(.,t)-v(.,t)||_(L∞(R^n))of the difference u-v decays faster than that of either u or v.This provides another example of the diffusion phenomenon of nonlinear hyperbolic waves,first observed by Hsiao,L.and Liu Taiping(see[1,2]).展开更多
The author proves a global existence result for strong solutions to the quasilinear dissipative hyperbolic equation (1.1) below, corresponding to initial values and source terms of arbitrary size, provided that the hy...The author proves a global existence result for strong solutions to the quasilinear dissipative hyperbolic equation (1.1) below, corresponding to initial values and source terms of arbitrary size, provided that the hyperbolicity parameter ε is sufficiently small. This implies a corresponding global existence result for the reduced quasilinear parabolic equation (1.4) below.展开更多
文摘The authors consider the asymptotic behavior of solutions of the quasilinear hyperbolic equation with linear damping u_(tt)+u_(t)-div(a(u)u)=0,and show that,at least when n≥3,they tend,as t-+∞,to those of the nonlinear parabolic equation v_t-div(a(v)v)=0,in the sense that the norm||u(.,t)-v(.,t)||_(L∞(R^n))of the difference u-v decays faster than that of either u or v.This provides another example of the diffusion phenomenon of nonlinear hyperbolic waves,first observed by Hsiao,L.and Liu Taiping(see[1,2]).
基金supported by the Fulbright Foundation (Chile, 2006)
文摘The author proves a global existence result for strong solutions to the quasilinear dissipative hyperbolic equation (1.1) below, corresponding to initial values and source terms of arbitrary size, provided that the hyperbolicity parameter ε is sufficiently small. This implies a corresponding global existence result for the reduced quasilinear parabolic equation (1.4) below.
