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On qualitative theory of solutions to nonlinear partial differential equations. / Mikhail Surnachev

Swansea University Author: Mikhail Surnachev

Abstract

"In this work I study certain aspects of qualitative behaviour of solutions to nonlinear PDEs. The thesis consists of introduction and three parts. In the first part I study solutions of Emden-Fowler type elliptic equations in nondivergence form. In this part I establish the following results;...

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Published: 2010
Institution: Swansea University
Degree level: Doctoral
Degree name: Ph.D
URI: https://cronfa.swan.ac.uk/Record/cronfa42611
Abstract: "In this work I study certain aspects of qualitative behaviour of solutions to nonlinear PDEs. The thesis consists of introduction and three parts. In the first part I study solutions of Emden-Fowler type elliptic equations in nondivergence form. In this part I establish the following results; 1. Asymptotic representation of solutions in conical domains; 2. A priori estimates for solutions to equations with weighted absorption term; 3. Existence and nonexistence of positive solutions to equations with source term in conical domains. In the second part I study regularity properties of nonlinear degenerate parabolic equations. There are two results here: A Harnack inequality and the H51der continuity for solutions of weighted degenerate parabolic equations with a time-independent weight from a suitable Muckenhoupt class; A new proof of the Holder continuity of solutions. The third part is propedeutic. In this part I gathered some facts and simple proofs relating to the Harnack inequality for elliptic equations. Both divergent and nondivergent case are considered. The material of this chapter is not new, but it is not very easy to find it in the literature. This chapter is built entirely upon the so-called ''growth lemma" ideology (introduced by E.M. Landis)."
Keywords: Theoretical mathematics.
College: Faculty of Science and Engineering

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