Approximate inertial manifolds for nonlinear parabolic equations via steady-state determining mapping. (English) Zbl 0864.35056
Summary: For nonlinear parabolic evolution equations, it is proved that, under the assumptions of local Lipschitz continuity of the nonlinearity and the dissipativity of semiflows, there exist approximate inertial manifolds (AIM) in the energy space and that the approximate inertial manifolds are constructed as the graph of the steady-state determining mapping based on the spectral decomposition. It is also shown that the thickness of the exponentially attracting neighborhood of the AIM converges to zero at a fractional power rate as the dimension of the AIM increases. Applications of the obtained results to Burgers’ equation, higher-dimensional reaction-diffusion equations, 2D Ginzburg-Landau equations, and axially symmetric Kuramoto-Sivashinsky equations in annular domains are included.
MSC:
35K60 | Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations |
35B40 | Asymptotic behavior of solutions to PDEs |
34G20 | Nonlinear differential equations in abstract spaces |
35G10 | Initial value problems for linear higher-order PDEs |
35K25 | Higher-order parabolic equations |