Qualitative Analysis Of Solutions For Several Classes Of Nonlinear Parabolic Equations And Keller-segel Chemotaxis Model

This thesis is devoted to the study of several classes of nonlinear parabolicequations (systems) came from physics, chemistry, biology and other applied science.From the second chapter to the fourth chapter, we discuss localization, global blow-up,single point blow-up, complete blow-up, Fujita critical exponent, blow-up rate,secondary critical exponent, large time asymptotic behavior, life span, finite-timeextinction and decay estimates for some kinds of nonlinear parabolic equations(systems). In the fifth chapter, we discuss the global boundedness and finite-timeblow-up for a quasilinear parabolic-elliptic Keller-Segel chemotaxis model withLogistic source. For the details, we divide the dissertation into five chapters:In Chapter1, we introduce the background and the development of the relatedtopics and summarize the main content of the present thesis.In Chapter2, we study Cauchy problem of a doubly degenerate parabolic equationwith a strongly nonlinear source. Firstly, by using the self-similar method, iterativemethod, comparison principle and parabolic regularization method, we obtain theproperties of localization of solutions for the problem. Secondly, under some suitableconditions, based on the contradiction argument, we prove that the solution of theCauchy problem blows up at any point of the whole space to arbitrary initial data withcompact support. Thirdly, under some suitable assumptions, we investigate single pointblow-up for a large of radial decreasing solutions of the Cauchy problem and give upperbound of the radial solution in a small neighborhood of the blow-up point. Finally,under some special conditions, by using the intersection comparison and self-similarmethod, we obtain the result of complete blow-up and the stability of complete blow-uptime with regard to the initial data.In Chapter3, we first discuss Cauchy problem for the non-Newtonian polytropicfiltration equation with a localized reaction. By constructing various self-similarsuper-sub solutions and energy method, we give the critical global existence exponentand critical Fujita exponent. Moreover, under some additional conditions, we obtain theresults of blow-up rate and blow-up set for the non-global solution. Then, we considerCauchy problem for a degenerate parabolic equation with inhomogeneous density andsource. By self-similar method and convex method, we obtain the secondary criticalexponent, the large time asymptotic behavior of the global solution and the precise estimate of life span for the blow-up solution.In Chapter4, we first consider the initial-boundary value problem for a polytropicfiltration equation with the nonlocal source and interior absorption. By usingLp-integral norm estimate method, Gagliardo-Nirenberg inequality and comparisonprinciple, we investigate the extinction property of solutions and give the decayestimates of extinction solutions. Then, we discuss the initial-boundary value problemfor a quasilinear parabolic system with nonlocal sources. By using energy method andcomparison principle, we obtain the sufficient conditions of finite-time extinctionsolutions.In Chapter5, we study a quasilinear parabolic-elliptic Keller-Segel chemotaxissystem with logistic source and homogeneous Neumann boundary conditions. Firstly,by using the standard Moser-Alikakos iteration, we obtain the global boundedness forthe problem. Secondly, by applying energy method, we give the sufficient conditions offinite-time blow-up of solutions.