Global existence and asymptotic behavior for a quasilinear reaction-diffusion system from climate modeling. (English) Zbl 0758.35044
Summary: This paper is concerned with a weakly coupled system of quasilinear autonomous strongly parabolic equations on a compact two-dimensional manifold without boundary; the system arises from an energy balance climate model. We establish \(L^ \infty\), Hölder, and Sobolev estimates, and apply general results on quasilinear evolution equations in order to guarantee the existence of classical nonnegative solutions. More precisely, it is shown that the system generates a global solution semiflow in the positive cone of some fractional order Sobolev space. Employing elements of the theory of infinite-dimensional dissipative systems, we prove the existence of a connected global attractor. Finally, we present some results about stationary solutions and forced periodic oscillations. The present paper extends earlier work of the authors on a semilinear problem [Nonlinear Anal., Theory Methods Appl. 14, No. 11, 915-926 (1990; Zbl 0709.35055)].
MSC:
| 35K57 | Reaction-diffusion equations |
| 35K55 | Nonlinear parabolic equations |
| 86A10 | Meteorology and atmospheric physics |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 58J60 | Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.) |
| 58E07 | Variational problems in abstract bifurcation theory in infinite-dimensional spaces |
| 35G10 | Initial value problems for linear higher-order PDEs |
| 35K25 | Higher-order parabolic equations |
Keywords:
two-dimensional connected compact oriented Riemannian manifold; weakly coupled system of quasilinear autonomous strongly parabolic equations; manifold without boundary; energy balance climate model; Hölder estimates; Sobolev estimates; quasilinear evolution equations; existence of classical nonnegative solutions; fractional order Sobolev space; theory of infinite-dimensional dissipative systems; existence of a connected global attractor; stationary solutions; forced periodic oscillationsCitations:
Zbl 0709.35055References:
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