It is known that the one-dimensional nonlinear heat equation ut = f(u)x1x1,f'(u) > 0,u(±∞,t) = u±,u+ = u_ has a unique self-similar solution u(x1/1+t).In multi-dimensional space,u(x1/1+t) is called a...It is known that the one-dimensional nonlinear heat equation ut = f(u)x1x1,f'(u) > 0,u(±∞,t) = u±,u+ = u_ has a unique self-similar solution u(x1/1+t).In multi-dimensional space,u(x1/1+t) is called a planar diffusion wave.In the first part of the present paper,it is shown that under some smallness conditions,such a planar diffusion wave is nonlinearly stable for the nonlinear heat equation:ut-△f(u) = 0,x ∈ Rn.The optimal time decay rate is obtained.In the second part of this paper,it is further shown that this planar diffusion wave is still nonlinearly stable for the quasilinear wave equation with damping:utt + utt+ △f(u) = 0,x ∈ Rn.The time decay rate is also obtained.The proofs are given by an elementary energy method.展开更多
This paper is concerned with the existence and the nonlinear asymptotic stabil- ity of traveling wave solutions to the Cauchy problem for a system of dissipative evolution equations {θt=vζx＋（ζθ）x＋aθxx,ζt=-θ...This paper is concerned with the existence and the nonlinear asymptotic stabil- ity of traveling wave solutions to the Cauchy problem for a system of dissipative evolution equations {θt=vζx＋（ζθ）x＋aθxx,ζt=-θx＋βζxx;with initial data and end states （ζθ）（x,0）=（ζ0,θ0）（x）→（ζ±,θ±）as x→∞.We obtain the existence of traveling wave solutions by phase plane analysis and show the asymptotic nonlinear stability of traveling wave solutions without restrictions on the coeffi- cients a and v by the method of energy estimates.展开更多
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）...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.展开更多
The nonlinear evolution problem in nonparallel boundary layer stability was studied. The relative parabolized stability equations of nonlinear nonparallel boundary layer were derived. The developed numerical method, w...The nonlinear evolution problem in nonparallel boundary layer stability was studied. The relative parabolized stability equations of nonlinear nonparallel boundary layer were derived. The developed numerical method, which is very effective, was used to study the nonlinear evolution. of T-S disturbance wave at finite amplitudes. Solving nonlinear equations of different modes by using predictor-corrector and iterative approach, which is uncoupled between modes, improving computational accuracy by using high order compact differential scheme, satisfying normalization condition I determining tables of nonlinear terms at different modes, and implementing stably the spatial marching, were included in this method. With different initial amplitudes, the nonlinear evolution of T-S wave was studied. The nonlinear nonparallel results of examples compare with data of direct numerical simulations (DNS) using full Navier-Stokes equations.展开更多
基金supported in part by National Basic Research Program of China (GrantNo. 2006CB805902)supported in part by National Natural Science Foundation of China for Distinguished Youth Scholar (Grant No. 10825102)+1 种基金NSFC-NSAF (Grant No. 10676037)National Basic Research Program of China (Grant No. 2006CB805902)
文摘It is known that the one-dimensional nonlinear heat equation ut = f(u)x1x1,f'(u) > 0,u(±∞,t) = u±,u+ = u_ has a unique self-similar solution u(x1/1+t).In multi-dimensional space,u(x1/1+t) is called a planar diffusion wave.In the first part of the present paper,it is shown that under some smallness conditions,such a planar diffusion wave is nonlinearly stable for the nonlinear heat equation:ut-△f(u) = 0,x ∈ Rn.The optimal time decay rate is obtained.In the second part of this paper,it is further shown that this planar diffusion wave is still nonlinearly stable for the quasilinear wave equation with damping:utt + utt+ △f(u) = 0,x ∈ Rn.The time decay rate is also obtained.The proofs are given by an elementary energy method.
基金Acknowledgements He＇s research is supported in part by National Basic Research Program of China （Grant No. 2006CB805902）. Huang＇ research is supported in part by National Natural Science Foundation of China for Distinguished Youth Scholar （Grant No. 10825102）, NSFC-NSAF （Grant No. 10676037） and National Basic Research Program of China （Grant No. 2006CB805902）.
基金supported by the Natural Science Foundation of China(11001095)the Ph.D.specialized grant of the Ministry of Education of China(20100144110001)+2 种基金the Special Fund for Basic Scientific Research of Central Colleges(CCNU12C01001)supported by the Fundamental Research Funds for the Central Universities(2015IA009)the Natural Science Foundation of China(61573012)
文摘This paper is concerned with the existence and the nonlinear asymptotic stabil- ity of traveling wave solutions to the Cauchy problem for a system of dissipative evolution equations {θt=vζx＋（ζθ）x＋aθxx,ζt=-θx＋βζxx;with initial data and end states （ζθ）（x,0）=（ζ0,θ0）（x）→（ζ±,θ±）as x→∞.We obtain the existence of traveling wave solutions by phase plane analysis and show the asymptotic nonlinear stability of traveling wave solutions without restrictions on the coeffi- cients a and v by the method of energy estimates.
基金Supported by the National Natural Science Foundation of China(No.11201301)Shanghai University Young Teacher Training Program(No.slg12026)
文摘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.
文摘The nonlinear evolution problem in nonparallel boundary layer stability was studied. The relative parabolized stability equations of nonlinear nonparallel boundary layer were derived. The developed numerical method, which is very effective, was used to study the nonlinear evolution. of T-S disturbance wave at finite amplitudes. Solving nonlinear equations of different modes by using predictor-corrector and iterative approach, which is uncoupled between modes, improving computational accuracy by using high order compact differential scheme, satisfying normalization condition I determining tables of nonlinear terms at different modes, and implementing stably the spatial marching, were included in this method. With different initial amplitudes, the nonlinear evolution of T-S wave was studied. The nonlinear nonparallel results of examples compare with data of direct numerical simulations (DNS) using full Navier-Stokes equations.
