Abstract
We are concerned with the asymptotic dynamics of a certain type of semilinear parabolic equation, namely,u t=u xx+(f(u))x+g(u)+h(x) on the interval [0,L]. Under the general condition we prove that this equation admits a dissipative dynamical system and it possesses the global attractor. But for largeL > 0, we do not know whether or not an inertial manifold exists. Here we introduce a nonlinear change of variables so that we transform the above equation to a reaction diffusion system which possesses exactly the same asymptotic dynamics. We then prove the existence of an inertial manifold for the transformed equation; thereby we find the ordinary differential equation which describes completely the long-time dynamics of the orginal equation.
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Kwak, M. Finite-dimensional description of convective Reaction-Diffusion equations. J Dyn Diff Equat 4, 515–543 (1992). https://doi.org/10.1007/BF01053808
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DOI: https://doi.org/10.1007/BF01053808