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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)

MSC:

35J70 Degenerate elliptic equations
35K65 Degenerate parabolic equations
35R70 PDEs with multivalued right-hand sides
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