Asymptotic Stability Of Solutions To The Hamilton-Jacobi Equation

By means of energy law and an important inequality,the study of the Cauchy problem of nonlinear development equation,we obtained a asymptotic stability-related to the solution of Hamilton-Jacobi equation ut + f(Du)= ?u in a peace-keeping multidimensional space.The initial data of the equation ut+f(Du)= ?u is prescribed as u(x,0)= u0(x)?u±(u+?u-),x1 ?±?.In the one-dimensional case,we prove that under some smallness conditions,the solutions to ut + f(Du)= ?u converge to a diffusion wave u(x1/(?))as t tends to +?,where u(x1/(?))is the unique self-similar solution to the one-dimensional nonlinear equation where CO = 1/2f"(?1)|?1=0.The convergence rate in time is obtained.We further prove that for multidimensional case,the solutions to ut+f(Du)=?u still converge to u(x1/(?))with C0 =1/2(?)2f(?)/(?)?12|?=0=0,?=(?1,?2,…?n)under some smallness conditions,where t(x1/(?))is called planar diffusion wave in mul-tidimensional case and the time decay rate is also obtained.