Stability w.r.t. Disturbances for the Global Attractor of Multi-Valued Semiflow Generated by Nonlinear Wave Equation (original) (raw)
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Dedicated to Hiroshi Matano on the occasion of his 60th birthday ABSTRACT. In this note we review the characterization of global attractors for dissipative semiflows generated by scalar semilinear parabolic equations of the form ut=uxx+f(x,u−ux)u_{t}=u_{xx}+f(x, u-u_{x})ut=uxx+f(x,u−ux) defined on the interval 0leqxleqpi0\leq x\leq\pi0leqxleqpi with Neumann boundary conditions. We outline the characterization results for these global attractors-the Sturm attractors-obtained by a permutation of the equilibrium solutions-the Sturm permutation-associated to the second order ODEODEODE satisfied by the stationary solutions of the parabolic equation. In particular we consider the characterization results for the class of nonlinearities of the form f=f(u)f=f(u)f=f(u)-the Hamiltonian class. Using this characterization we then outline some results on the geometry of Sturm attractors for the case of periodic boundary conditions.