Khrabustovskyi A. Homogenization of spectral and evolution problems on Riemannian manifolds of complex microstructure

The objects of investigation: the spectrum of Laplace-Beltrami operator, Cauchy problem for the wave equation, the reaction-diffusion systems modelling the processes of diffusion and reaction of particles of several types. The aims of investigation: homogenization of the spectrum of Laplace-Beltrami operator on Riemannian manifolds of complex microstructure, homogenization of the wave equation on pseudo-Riemannian manifolds of complex microstructure, the proof of maximum principle, conservation of positivity and stabilization of solutions for reaction-diffusion systems. Methods of investigations: variational methods of homogenization theory, methods of functional analysis and complex variable theory, methods of spectral theory of operators in Hilbert spaces. The obtained results: 1) The asymptotic behaviour of Laplace-Beltrami operator spectrum on Riemannian manifolds of complex microstructure of three qualitatively different types is described. For all types the homogenized operator is found; 2) The result of homogenization of Cauchy problem for wave equation on a pseudo-Riemannian manifold with a special metric, whose coefficients grow at a part of the manifold, is obtained. It is shown that as a result of homogenization the potential term appears in the equation; 3) Using methods of homogenization theory on manifolds the maximum principle, conservation of positivity, stabilization of solutions to a constant (which is found in explicit form) are proved for the reaction-diffusion systems. All results of the thesis are new. The results are theoretical and can be used to study other homogenization problems on Riemannian manifolds. The methodology used to study the properties of reaction-diffusion systems can be applied for qualitative analysis of other equations of mathematical physics.