Doubly nonlinear parabolic-type equations as dynamical systems. (English) Zbl 0802.35011
Summary: We study a class of doubly nonlinear parabolic PDEs, where, in addition to some weak nonlinearities, also mild nonlinearities of porous media type are allowed inside the time derivative. In order to formulate the equations as dynamical systems, some existence and uniqueness results are proved. Then the existence of a compact attractor is shown for a class of nonlinear PDEs that include doubly nonlinear porous medium-type equations. Under stronger smoothness assumptions on the nonlinearities, the finiteness of the fractal dimension of the attractor is also obtained.
MSC:
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35K55 | Nonlinear parabolic equations |
| 35K57 | Reaction-diffusion equations |
| 35Q40 | PDEs in connection with quantum mechanics |
| 76S05 | Flows in porous media; filtration; seepage |
Keywords:
doubly nonlinear parabolic PDEs; nonlinearities of porous media type; compact attractor; fractal dimensionReferences:
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