Nonstationary Problems and Spectral Problems
In this chapter, the results obtained and the methods developed in the preceding chapters will be applied to the study of the asymptotic behavior, as s → ∞, of solutions of initial boundary value problems in strongly perforated domains ω(s) (nonstationary problems) as well as to the study the, asymptotic behavior of eigenvalues and eigenfunctions of boundary value problems in such domains. This becomes possible since the obtained results can be interpreted as the strong convergence of resolvents of self-adjoint operators generated by these boundary value problems. This allows us to study the convergence of families of spectral projections for such operators and then to apply the Fourier method to solving nonstationary problems in cylindrical domains ω T (s) =ω(s)×[0, T]. In noncylindrical domains, the resolvent convergence is used directly. For concreteness, we will consider initial boundary value problems for the heat equation.
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