In this paper,we discuss a class of the quasillinear hyperbolic equations with the inhomogeneous terms: u_■+σ(v)+2α(t)u=0.v_■-u-0 Under the certain of hypothesis.we prove the globally existence theorems of the smo...In this paper,we discuss a class of the quasillinear hyperbolic equations with the inhomogeneous terms: u_■+σ(v)+2α(t)u=0.v_■-u-0 Under the certain of hypothesis.we prove the globally existence theorems of the smooth solutions for its Cauchy problem.展开更多
In this paper,we investigate group-invariant solutions to the hyperbolic geometric flow on Riemann surfaces,which include solutions of separation variables,traveling wave solutions,self-similar solutions and radial so...In this paper,we investigate group-invariant solutions to the hyperbolic geometric flow on Riemann surfaces,which include solutions of separation variables,traveling wave solutions,self-similar solutions and radial solutions.In the proceeding of reduction,there are elliptic,hyperbolic and mixed types of equations.For the first kind of equation,some exact solutions are found;while for the last two kinds,with implicit solutions found,we furthermore investigate whether there will be a global solution or blowing up.Referring to the work of Kong et al.(2009),the results come out perfectly.展开更多
Decay of the energy for the Cauchy problem of the wave equation of variable coefficients with a dissipation is considered. It is shown that whether a dissipation can be localized near infinity depends on the curvature...Decay of the energy for the Cauchy problem of the wave equation of variable coefficients with a dissipation is considered. It is shown that whether a dissipation can be localized near infinity depends on the curvature properties of a Riemannian metric given by the variable coefficients. In particular, some criteria on curvature of the Riemannian manifold for a dissipation to be localized are given.展开更多
This paper considers the exponential decay of the solution to a damped semilinear wave equation with variable coefficients in the principal part by Riemannian multiplier method. A differential geometric condition that...This paper considers the exponential decay of the solution to a damped semilinear wave equation with variable coefficients in the principal part by Riemannian multiplier method. A differential geometric condition that ensures the exponential decay is obtained.展开更多
For a compact Riemannian manifold NRK without boundary, we establish the existence of strong solutions to the heat flow for harmonic maps from Rn to N, and the regularizing rate estimate of the strong solutions. Moreo...For a compact Riemannian manifold NRK without boundary, we establish the existence of strong solutions to the heat flow for harmonic maps from Rn to N, and the regularizing rate estimate of the strong solutions. Moreover, we obtain the analyticity in spatial variables of the solutions. The uniqueness of the mild solutions in C([0,T]; W1,n) is also considered in this paper.展开更多
文摘In this paper,we discuss a class of the quasillinear hyperbolic equations with the inhomogeneous terms: u_■+σ(v)+2α(t)u=0.v_■-u-0 Under the certain of hypothesis.we prove the globally existence theorems of the smooth solutions for its Cauchy problem.
文摘In this paper,we investigate group-invariant solutions to the hyperbolic geometric flow on Riemann surfaces,which include solutions of separation variables,traveling wave solutions,self-similar solutions and radial solutions.In the proceeding of reduction,there are elliptic,hyperbolic and mixed types of equations.For the first kind of equation,some exact solutions are found;while for the last two kinds,with implicit solutions found,we furthermore investigate whether there will be a global solution or blowing up.Referring to the work of Kong et al.(2009),the results come out perfectly.
基金supported by the National Natural Science Foundation of China (Nos.60225003,60821091,10831007,60774025)KJCX3-SYW-S01
文摘Decay of the energy for the Cauchy problem of the wave equation of variable coefficients with a dissipation is considered. It is shown that whether a dissipation can be localized near infinity depends on the curvature properties of a Riemannian metric given by the variable coefficients. In particular, some criteria on curvature of the Riemannian manifold for a dissipation to be localized are given.
基金Project supported by the National Natural Science Foundation of China (No.60334040, No.60225003).
文摘This paper considers the exponential decay of the solution to a damped semilinear wave equation with variable coefficients in the principal part by Riemannian multiplier method. A differential geometric condition that ensures the exponential decay is obtained.
基金supported by National Natural Science Foundation of China (GrantNos. 10701064, 10931001 and 11171061) Fundamental Research Funds for the Central Universities Program(Grant Nos. 2012QNA3008 and 12D10912)
文摘For a compact Riemannian manifold NRK without boundary, we establish the existence of strong solutions to the heat flow for harmonic maps from Rn to N, and the regularizing rate estimate of the strong solutions. Moreover, we obtain the analyticity in spatial variables of the solutions. The uniqueness of the mild solutions in C([0,T]; W1,n) is also considered in this paper.
