On stable stationary solutions to a quasilinear parabolic equation

The asymptotic behavior of solutions to boundary value problems for quasilinear autono¬mous parabolic equations is studied. Let $S_1$ denote the set of stationary solutions $\varphi(x)$ to the problem which possess the following property: the spectral problem produced by the elliptic operator linearized on $\varphi(x)$ has at most one eigenvalue in the right halfplane of the complex plane. Also, assume that the nonlinear terms of the boundary value problem depend analytically on the unknown function and its derivatives. It is proved that either the set $S_1$ consists of isolated stationary solutions, or $S_1$ is a connected unbounded ordered family of stationary solutions. Let $S_1$ consist of isolated stationary solutions, and $\psi(x)$ be a nonstable stationary solution in $S_1$. It is proved that the stable manifold $W^S(\psi)$ divides the set of initial data into two components converging to different stationary solutions as $t\to+\infty$.