Stabilization of solutions of nonlinear and degenerate evolution equations.

*(English)*Zbl 0583.35059The present paper deals with the quasilinear parabolic problem
\[
u_ t=\Delta \eta (u)+f(x,u)\quad in\quad \Omega \times R^+,\quad u(x,0)=u_ 0(x)\quad in\quad \Omega,\quad u=u_ 1\quad in\quad \partial \Omega \times R^+,
\]
where \(\eta\) is a continuous, strictly increasing function with \(\eta (0)=0\); degenerate diffusion at \(u=0\) (namely \(\eta '(0)=0)\) is allowed. The authors prove a stabilization result, whose basic steps are as follows:

(a) Theorem 1.1: It is proved that if certain regularity conditions on the solution are satisfied ((0.6)), then any element in the omega-limit set of any (precompact) trajectory is a stationary solution (the omega- limit is understood here in terms of \(L^ 2\) convergence);

(b) Theorems 2.1, 2.4: They give conditions ensuring the validity of (0.6) for bounded (weak) solutions;

(c) Theorems 2.5, 2.6: They improve item (b), showing that under the same conditions convergence holds in the uniform norm;

(d) Examples and applications to the porous media equation and variations thereof.

The proofs of (a) rely on a direct calculation and on an appropriate choice of the test-functions. Step (b) is based on a priori estimates for u and its derivatives. Step (c) is established leaning on an interesting compactness result due to Di Benedetto; the examples and applications are worked out using comparison arguments and suitable super- and subsolutions. Related results had been proved by A. Schiaffino, A. Tesei and the reviewer [Ann. Mat. Pura Appl., IV. Ser. 136, 35- 48 (1984; Zbl 0556.35083)].

(a) Theorem 1.1: It is proved that if certain regularity conditions on the solution are satisfied ((0.6)), then any element in the omega-limit set of any (precompact) trajectory is a stationary solution (the omega- limit is understood here in terms of \(L^ 2\) convergence);

(b) Theorems 2.1, 2.4: They give conditions ensuring the validity of (0.6) for bounded (weak) solutions;

(c) Theorems 2.5, 2.6: They improve item (b), showing that under the same conditions convergence holds in the uniform norm;

(d) Examples and applications to the porous media equation and variations thereof.

The proofs of (a) rely on a direct calculation and on an appropriate choice of the test-functions. Step (b) is based on a priori estimates for u and its derivatives. Step (c) is established leaning on an interesting compactness result due to Di Benedetto; the examples and applications are worked out using comparison arguments and suitable super- and subsolutions. Related results had been proved by A. Schiaffino, A. Tesei and the reviewer [Ann. Mat. Pura Appl., IV. Ser. 136, 35- 48 (1984; Zbl 0556.35083)].

Reviewer: P.de Mottoni

##### MSC:

35K55 | Nonlinear parabolic equations |

35B35 | Stability in context of PDEs |

35B40 | Asymptotic behavior of solutions to PDEs |

35B30 | Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs |

##### Keywords:

stability; asymptotic behaviour; quasilinear parabolic problem; degenerate diffusion; stabilization
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\textit{M. Langlais} and \textit{D. Phillips}, Nonlinear Anal., Theory Methods Appl. 9, 321--333 (1985; Zbl 0583.35059)

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