带对流项的一维渗流方程解的渐近性质

LARGE-TIME BEHAVIOUR OF SOLUTIONS OF THE POROUS MEDIUM EQUATION WITH CONVECTION

如果我们考虑不可压缩流体在均匀刚性多孔介质中的非饱和流动,在适当前提下,垂直方向流体的运动就由方程(1.1)所描述。此处(φ(u))_(xx)和(g(u))_x分别为扩散项和对流项,φ和g通常由流体和多孔介质的特性所决定。

We consider the large-time behaviour of solutions of the Cauchy problem and the Cauchy-Dirichlet problem for the nonlinear degenerate parabolic equation u_t=(φ(u))_(xx)-(g(u))_x.We prove that the large-time behaviour of solutions depends strongly upon the shape of g(convex or concave) as well as upon the values u^-=u_o(-∞) and u^+=u_o(+∞).Furthermore, we give sufficient conditions under which the solution converges to a travelling wave and converges to a rarefaction wave respectively;moreover,the rate of convergence to a rarefaction wave is stimated.