Cirstea, Florica-Corina (2004) Nonlinear Methods in the Study of Singular Partial Differential Equations. PhD thesis, Victoria University.

Abstract

Nonlinear singular partial differential equations arise naturally when studying models from such areas as Riemannian geometry, applied probability, mathematical physics and biology. The purpose of this thesis is to develop analytical methods to investigate a large class of nonlinear elliptic PDEs underlying models from physical and biological sciences. These methods advance the knowledge of qualitative properties of the solutions to equations of the form &Delta u= &fnof(x,u) where &Omega is a smooth domain in R^N (bounded or possibly unbounded) with compact (possibly empty) boundary &part&Omega. A non-negative solution of the above equation subject to the singular boundary condition u(x)&rarr &infin as dist(x,&part&Omega)&rarr 0 (if &Omega&ne R^N), or u(x)&rarr &infin as | x | &rarr &infin (if &Omega=R^N) is called a blow-up or large solution; in the latter case the solution is called an entire large solution. Issues such as existence, uniqueness and asymptotic behavior of blow-up solutions are the main questions addressed and resolved in this dissertation. The study of similar equations with homogeneous Dirichlet boundary conditions, along with that of ODEs, supplies basic tools for the theory of blow-up. The treatment is based on devices used in Nonlinear Analysis such as the maximum principle and the method of sub and super-solutions, which is one of the main tools for finding solutions to boundary value problems. The existence of blow-up solutions is examined not only for semilinear elliptic equations, but also for systems of elliptic equations in R^N and for singular mixed boundary value problems. Such a study is motivated by applications in various fields and stimulated by very recent trends in research at the international level. The influence of the nonlinear term &fnof(x,u) on the uniqueness and asymptotics of the blow-up solution is very delicate and still eludes researchers, despite a very extensive literature on the subject. This challenge is met in a general setting capable of modelling competition near the boundary (that is, 0&sdot &infin near &part &Omega), which is very suitable to applications in population dynamics. As a special feature, we develop innovative methods linking, for the first time, the topic of blow-up in PDEs with regular variation theory (or Karamata's theory) arising in applied probability. This interplay between PDEs and probability theory plays a crucial role in proving the uniqueness of the blow-up solution in a setting that removes previous restrictions imposed in the literature. Moreover, we unveil the intricate pattern of the blow-up solution near the boundary by establishing the two-term asymptotic expansion of the solution and its variation speed (in terms of Karamata's theory). The study of singular phenomena is significant because computer modelling is usually inefficient in the presence of singularities or fast oscillation of functions. Using the asymptotic methods developed by this thesis one can find the appropriate functions modelling the singular phenomenon. The research outcomes prove to be of significance through their potential applications in population dynamics, Riemannian geometry and mathematical physics.

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