Asymptotic Behavior Analysis To Solutions Of Several Nonlinear Diffusion Equations

Avariety diffusion phenomena appear wildly in nature. For instance, there are a lotof diffusion phenomenon in combustion, phase transition, biochemistry and biologicalgroups. These diffusion phenomena can be modelled by partial differential equationswith diffusion terms. It is well known that the nonlinear equation is more authentic thanthe linear equation to describe or reflect some actual phenomenon. So the researchabout nonlinear diffusion equations is particularly important. In this thesis we mainlyanalyse the asymptotic behavior of solutions to several diffusion equations arose inapplied sciences. We divide the dissertation into six chapters:In Chapter1, we first introduce the background and development of the problemsin this paper, and then state the main content of this thesis.In Chapter2, we investigate Cauchy problems of two nonlinear diffusion equationswith variable coefficient. We first deal with the fast diffusion equation with variablecoefficient, and get a secondary critical exponent and life span estimate of blow-upsolutions by some inequality skills, employing ordinal differential equations andconstructing special functions. Then we consider the non-Newtonial diffusion equationwith variable coefficient, base on asymptotic behavior of fundamental solution top-Laplacian equation and scale transformation, we get the critical Fujita exponent,furthermore establish the secondary critical exponent on the decay asymptotic behaviorof an initial value at infinity.In Chapter3, we study initial-boundary value problem of two nonlinear diffusionequations with multiple nonlinearities. We first consider a non-Newtonial diffusionequation with a source and nonlinear boundary flux, and obtain the critical globalexistence curve and critical Fujita curve by constructing various self-similarsupersolutions and subsolutions. Then we deals with a fast diffusion system with innerabsorption and coupled through boundary. As the results of the interaction among themulti-nonlinearities in the system, we established the critical global existence curve byconstructing various super-solutions and sub-solutions, which are simply described via apair of parameters solving the characteristic algebraic system.In Chapter4, we study initial-boundary value problems of two nonlinear diffusionequations in logarithmic form. We first consider a nonlinear diffusion equation withlogarithmic boundary flux, and obtain the critical global existence exponent and critical Fujita exponent by constructing various self-similar supersolutions and subsolutions,furthermore give the blow-up rate for the nonglobal solutions. Then we study anonlinear diffusion equation with inner absorption and logarithmic boundary flux,establish the critical global existence curve and give the asymptotic behaviour close tothe blow-up time.In Chapter5, we deal with coupled nonlinear diffusion equations with absorptions.We characterize the range of parameters for which non-simultaneous blow up occurs,and establish the necessary and sufficient conditions for the occurrence ofnon-simultaneous blow-up with proper initial data. Moreover, we obtain the optimalcondition under which any blow-up is non-simultaneous.In Chapter6, we first find finite travelling-wave solutions, and then investigate theshort time development of interfaces for non-Newtonian diffusion equations with strongabsorption. We show that the initial behavior of the interface depends on theconcentration of the initial data. More precisely, we find a critical value of theconcentration, which separates the heating front of interfaces from the cooling front ofthem.