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Stabilization of solutions of nonlinear and degenerate evolution equations. (English) Zbl 0583.35059
The 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)].
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
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[1] Amann, H., On the existence of positive solutions of nonlinear elliptic boundary value problems, Indiana univ. math. J., 21, 125-146, (1971) · Zbl 0219.35037
[2] Aronson, D.; Crandall, M.G.; Peletier, L.A., Stabilization of solutions of a degenerate nonlinear problem, Nonlinear analysis, 6, 1001-1022, (1982) · Zbl 0518.35050
[3] Aronson, D.; Peletier, L.A., Large time behaviour of solutions of the porous medium equation in bounded domains, J. diff. eqns, 39, 378-412, (1981) · Zbl 0475.35059
[4] Bear, J., Dynamics of fluids in porous media, (1972), American Elsevier New York · Zbl 1191.76001
[5] Berryman, J.G.; Holland, C.J., Stability of the separable solution for fast diffusion, Archs ration. mech. analysis, 74, 379-399, (1980) · Zbl 0458.35046
[6] Bertsch, M., A class of degenerate diffusion equations with a singular nonlinear term, Nonlinear analysis, 7, 117-127, (1983) · Zbl 0509.35047
[7] Bertsch, M.; Nanbu, T.; Peletier, L.A., Decay of solutions of a degenerate nonlinear diffusion equation, Nonlinear analysis, 6, 539-554, (1982) · Zbl 0487.35051
[8] Chafee, N.; Infante, E., A bifurcation problem for a nonlinear parabolic equation, Applicable analysis, 4, 17-37, (1974) · Zbl 0296.35046
[9] Dafermos, C., Asymptotic behavior of solutions of evolution equations, () · Zbl 0499.35015
[10] Dibenedetto, E., Continuity of weak solutions to a general porous medium equation, Indiana univ. math. J., 32, 83-119, (1983) · Zbl 0526.35042
[11] D\sciB\scenedetto E., A boundary modulus of continuity for a class of singular parabolic equations, to appear.
[12] Fife, P.C.; Peletier, L.A., Nonlinear diffusion in population genetics, Archs ration. mech. analysis, 64, 93-109, (1977) · Zbl 0361.92020
[13] Fleming, W.H., A selection migration model in population genetics, J. math. biology, 2, 219-234, (1975) · Zbl 0325.92009
[14] Gurtin, M.E.; McCamy, On the diffusion of biological populations, Math. biosci., 33, 35-49, (1977) · Zbl 0362.92007
[15] Keller, H.B., Elliptic boundary value problems suggested by nonlinear diffusion processes, Archs ration. mech. analysis, 35, 363-381, (1969) · Zbl 0188.17102
[16] Ladyzenskaja, O.A.; Solonnikov, V.A.; Ural’ceva, N.N., Linear and quasilinear equations of parabolic type, ()
[17] Lions, P.L., On the existence of positive solutions of semilinear elliptic equations, SIAM rev., 24, 441-468, (1982)
[18] Matano, H., Asymptotic behavior and stability of solutions of semilinear diffusion equations, Publ. R.I.M.S. Kyoto univ., 15, 401-454, (1979) · Zbl 0445.35063
[19] Matano, H., Convergence of solutions of one dimensional semilinear parabolic equations, J. math. Kyoto univ., 18, 221-227, (1978) · Zbl 0387.35008
[20] Okubo, A., Diffusion and ecological problems. mathematical models, () · Zbl 0422.92025
[21] S\scacks, P.E., Global behaviour for a class of nonlinear evolution equations.
[22] Smoller, J.; Wasserman, J., Global bifurcation of steady state solutions, J. diff. eqns, 39, 269-290, (1981) · Zbl 0425.34028
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