Asymptotic Stability of Solutions to the Nonlinear Equation with End Sates of the Initial Data

This work describes the asymptotic stability of the solutions for equation
$u_{t}+f(\nabla u, u)=\Delta g(u)$
, where
$\lim\limits_{x_{1}\rightarrow\pm\infty, t=0}u(x, t)=u_{\pm}$
and
$u_{+}\neq u_{-}$
. The solution proved that under smallness conditions on initial data is expected to behave as the diffusion wave, which is the solution of one-dimensional case equation due to Darcy's law. The planar diffusion wave
$\overline{u}(\frac{x}{\sqrt{1+t}})$
have time-asymptotically behave as those of Darcy's law for the one-dimensional case equation, where
$\overline{u}(\frac{x}{\sqrt{1+t}})$
is solution to the one dimension equation
$\overline{u}_{t}+C_{0}\overline{u}_{x_{1}}^{2}=\overline{u}_{x_{1}x_{1}}, C_{0}= \frac{1}{2}\frac{\partial^{2}f(\xi)}{\partial\xi_{1}^{2}}\vert_{\xi=0},\xi=(\xi_{1},\xi_{2},\cdots,\xi_{n}),\overline{u}(\pm\infty, t)=u_{\pm}, u_{+}\neq u\_$
. At the same time, we get the convergence rate.