On the multiplicity of equilibrium solutions to a nonlinear diffusion equation on a manifold arising in climatology. (English) Zbl 0892.35065

The authors consider the stabilization (as \(t\to\infty\)) of solutions of \[ u_t-\Delta_pu+ {\mathcal G}(u)\in QS(x)\beta(u)+ f(t,x)\quad \text{in } (0,\infty)\times M, \qquad u(0,x)= u_0(x) \] on a two-dimensional compact oriented Riemannian manifold as well as the existence of multiple equilibria for certain \(Q\) in the following setting: \(\mathcal G\) denotes a continuous strictly increasing function on \(\mathbb{R}\) with \({\mathcal G}(0)=0\), \(Q\in\mathbb{R}\) is a crucial parameter, \(\beta\) is a bounded maximal monotone graph in \({\mathbb{R}}^2\), \(S\in L^\infty(M)\) and \(f\in L^\infty((0,\infty)\times M)\). The problem arises in the context of energy balance climate models. Stabilization of trajectories occurs under very general conditions in the sense that the \(L^2\)-\(\omega\)-limit of every sufficiently regular trajectory is contained in the set of stationary solutions. Moreover, this \(\omega\)-limit is compact in \(L_2\) and closed in \(W^{1,p}\). The existence of multiple stationary solutions is derived for a certain parameter range by the method of sub- and supersolutions.
Reviewer: G.Hetzer (Auburn)


35J70 Degenerate elliptic equations
35K65 Degenerate parabolic equations
35R70 PDEs with multivalued right-hand sides
Full Text: DOI


[1] Alt, H. W.; Luckhaus, S., Quasilinear elliptic-parabolic differential equations, Math. Z., 183, 311-341 (1983) · Zbl 0497.35049
[2] Amann, H., Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev., 18, 620-709 (1976) · Zbl 0345.47044
[4] Aubin, T., Nonlinear Analysis on Manifolds: Monge-Ampère Equations (1982), Springer Verlag · Zbl 0512.53044
[5] Benilan, Ph.; Crandall, M. G.; Sachs, P., Some \(L^1\), Appl. Math. Optim., 17, 203-224 (1988) · Zbl 0652.35043
[6] Brezis, H., Proprietés régularisantes de certains semi-groupes nonlinéaires, Israel J. Math., 9, 513-534 (1971) · Zbl 0213.14903
[7] Brezis, H., Opérateurs maximaux monotones et semigroupes de contractions dans les espaces de Hilbert (1973), North Holland: North Holland Amsterdam · Zbl 0252.47055
[8] Brezis, H.; Nirenberg, L., \(H^1C^1\), C. R. Acad. Sci. Paris, Série I, 317, 465-472 (1993) · Zbl 0803.35029
[9] Budyko, M. I., The effects of solar radiation variations on the climate of the Earth, Tellus, 21, 611-619 (1969)
[10] Dı́az, J. I., Mathematical analysis of some diffusive energy balance climate models, Mathematics, Climate and Environment (1993), Masson: Masson Paris, p. 28-56 · Zbl 0804.92026
[11] Dı́az, J. I.; Tello, L., Sobre un modelo bidimensional en Climatologı́a, (Casal, A., Actas del, XIII CEDYA/III Congreso de Matemática Aplicada (1995)), 310-315
[12] Dı́az, J. I.; Tello, L., A nonlinear parabolic problem on a Riemannian manifold without boundary arising in Climatology, Collectanea Mathematica (1997) · Zbl 0936.35095
[13] Dı́az, J. I.; de Thélin, F., On a nonlinear parabolic problem arising in some models related to turbulent flows, SIAM J. Math. Anal., 25, 1085-1111 (1994) · Zbl 0808.35066
[14] Hernández, J., Qualitative methods for nonlinear diffusion equations, (Fasano, A.; Primicerio, M., Nonlinear Diffusion Equations. Nonlinear Diffusion Equations, Lecture Notes (1986), Springer Verlag: Springer Verlag New York), 47-118 · Zbl 0626.35032
[15] Hetzer, G., The structure of the principal component for semilinear diffusion equations from energy balance climate models, Houston J. Math., 16, 203-216 (1990) · Zbl 0723.58051
[16] Hetzer, G., S-shapedness for energy balance climate models of Sellers type, (Dı́az, J. I., The Mathematics of Models for Climatology and Environment. The Mathematics of Models for Climatology and Environment, NATO ASI Series I: Global Environmental Change, 48 (1996), Springer Verlag: Springer Verlag Heidelberg), 253-288 · Zbl 0893.76003
[17] Nakao, M., A difference inequality and its application to nonlinear evolution equations,, J. Math. Soc. Japan, 30, 747-762 (1978) · Zbl 0388.35007
[18] North, G. R., Introduction to simple climate models, (Dı́az, J. I.; Lions, J. L., Mathematics, Climate and Environment (1993), Masson: Masson Paris), 139-159 · Zbl 0792.92028
[19] Ouyang, T., On the positive solutions of semilinear equations Δ \(uu hu^p =0\) on compact manifolds, Trans. Amer. Math. Soc., 331, 503-527 (1992) · Zbl 0759.35021
[20] Ouyang, T., On the positive solutions of semilinear equations Δ \(uu hu^p =0\) on compact manifolds, part II, Indiana Univ. Math. J., 40, 1083-1141 (1991) · Zbl 0773.35020
[21] Sellers, W. D., A global climatic model based on the energy balance of the earth- atmosphere system, J. Appl. Meteorol., 8, 392-400 (1969)
[22] Simon, J., Compact sets in the space \(L^p(0,TB\), Annali Mat. Pura Appl., CXLVI, 65-96 (1987) · Zbl 0629.46031
[23] Stone, P. H., A simplified radiative-dynamical model for the static stability of rotating atmospheres, J. Atmospheric Sci., 29, 405-418 (1972)
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