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On stable stationary solutions to a quasilinear parabolic equation. (English. Russian original) Zbl 0838.35063

Sib. Math. J. 34, No. 2, 233-241 (1993); translation from Sib. Mat. Zh. 34, No. 2, 45-51 (1993).
The paper deals with the asymptotic behavior of solutions to boundary value problems for quasilinear autonomous parabolic equations. Let \(S_1\) be 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, suppose that the nonlinear terms of the boundary value problem depend analytically on the unknown function and its derivatives. It is proved that either \(S_1\) consists of isolated stationary solutions, or \(S_1\) has a connected unbounded ordered family of stationary solutions. Assume that \(S_1\) consists of isolated stationary solutions, and \(\psi(x)\) is a non-stable stationary solution in \(S_1\). It is proved that the stable manifold of \(\psi(x)\), \(W^s(\psi)\) divides the set of initial data into two components converging to different stationary solutions as \(t\to +\infty\).

MSC:

35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
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