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Doubly Degenerate Parabolic Equations with Nonlinear Inner Sources or Boundary Flux

Author: JiangChaoXin
Tutor: ZhengSiNing
School: Dalian University of Technology
Course: Basic mathematics
Keywords: nonlinear doubly degenerate parabolic system nonlinear doubly degenerate parabolic inequality nonlinear inner source nonlinear boundary condition nonlinear inner absorption global existence blow-up critical exponent Liouville-type theorem
CLC: O175.26
Type: PhD thesis
Year: 2009
Downloads: 91
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Abstract


This thesis studies interactions among multi-nonlinearities and their influences to global existence and finite time blow-up of solutions in doubly degenerate parabolic equations (systems) with nonlinear source(absorption) and(or) boundary flux.This is a class of important nonlinear parabolic equations(systems) possessing wide applications in many fields such as filtration,phase transition,physics,chemistry,biochemistry and dynamics of biological groups,image processing.The double degeneracy as well as the nonlinearities from the source and(or) boundary flux make the studies more complicated and difficult.In general the doubly degenerate models considered here do not admit classical solutions, we have to start from weak solutions and related comparison principles.We are interested in the complicated interactions among multi-nonlinearities from double degeneracy and source and(or) flux.We will consider the existence and non-existence of global solutions by suitable comparison principle,and establish the critical exponents via precise and complete classifications for nonlinear parameters.In addition,we obtain a Liouville-type theorem for a doubly degenerate parabolic inequality by means of the test function method.This thesis is organized as follows.Chapter 1 is an introduction to recall the background and the current development of the related topics and to summarize the main results of the present thesis.Chapter 2 studies a doubly degenerate parabolic equation subject to nonlinear inner source and boundary flux.We discuss interaction among the four nonlinear mechanisms in the model,and establish the sufficient and necessary condition for global non-negative weak solutions by weak comparison principle.Chapter 3 considers a doubly degenerate parabolic equation subject to inner absorption and boundary flux.The weak comparison principle for weak solutions and a careful classification for the four nonlinear parameters give us the critical boundary exponent to describe the sufficient and necessary condition for global non-negative weak solutions of the model,where,in particular,a precise analysis is included to show the significant contribution of the absorption coefficient to the critical property of solutions. Chapter 4 at first concerns a doubly degenerate parabolic inequality.A Liouville-type theorem is obtained by carefully choosing desired test functions for weak solutions of the inequality problem.Next,critical global existence exponent and critical Fujita exponent are proved for the half space problem to a doubly degenerate parabolic equation with nonlinear flux condition.Chapter 5 focuses on a system of doubly degenerate parabolic equations coupled via inner source and boundary flux.Under the framework of weak solutions and weak comparison principle,a detail discussion is made for the interaction among the multi-nonlinearities. The critical exponent(i.e.the sufficient and necessary conditions for global weak solutions) is determined via a complete classification to all the eight nonlinear parameters included.

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CLC: > Mathematical sciences and chemical > Mathematics > Mathematical Analysis > Differential equations, integral equations > Partial Differential Equations > Parabolic equation
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