The invariant sets and the solutions of the 1+2-dimensional generalized thin film equation are discussed. It is shown that there exists a class of solutions to the equations, which are invariant with respect to the set $$E_0 = \{ u:u_x = v_x F(u),u_y = v_y F(u)\} ,$$ where v is a smooth function of variables x, y and F is a smooth function of u. This extends the results of Galaktionov (2001) and for the 1+1-dimensional nonlinear evolution equations.
Chang-zheng QU & Chun-rong ZHU Center for Nonlinear Studies, Northwest University, Xi’an 710069, China
In this paper, we establish a priori estimates to the generalized second order Toda system{-Δu1（x）=2R1（x）eu1-R2eu2,-Δu2（x）=R1（x）eu1＋2R2eu2 in R2 , and discuss the convergence and asymptotic behavior of its solutions, where Ri（x）, i=1, 2, is bounded function in R2 . Consequently, we prove that all the solutions satisfy an identity, which is somewhat a generalization of the well-known Kazdan-Warner condition.
