Qualitative Analysis For Several Types Of Reaction-Diffusion Problems

Reaction-diffusion equation(s) are usually used to describe some application problems arising in physics, chemistry, ecology and other disciplines. The study on the existence of solutions of various types and the underlying dynamics of such equations has always been an important research topic in differential equa-tion theory and application. This Ph. D. dissertation mainly studied four kinds of reaction-diffusion models:a stream population model under water action, a selection-migration model in population genetics, a population invasion model with free boundary and a combustion model with double free boundary fronts, in which the former two are fixed domain problems and the latter two are moving domain problems. By qualitative analysis, maximum principle, comparison principle, spec-tral theory, bifurcation theory and upper and lower solution method, we explore how the parameters appearing in the equations (say, diffusion rate, advection rate, habitat size, initial data, etc) affect the potential dynamical behaviors, and obtain different dynamics in different parameter ranges, which in turn reveal the essential effects of different environments on the practical problem. Theoretically we gener-alize or improve some previous work, and develop some new methods and skills. On the application side, our results explain some observed phenomena, and provide a basic theory for understanding the evolutionary mechanism of practical problems. The concrete research contents are divided into the following four sections:Organisms that inhabit aquatic ecosystems are carried downstream or even washed out by the unidirectional water flow, which consequently causes a decline or extinction in population. Despite this bias, population resist washout and manage to persist over many generations in such advective environments. This biological phenomenon is known as "drift paradox", and was observed as early as 1954 by Muller [123]. People have been committed to exploring the mechanism behind this phenomenon and resort to mathematical modeling. In 2001, Speirs, Gurney [145] used the following reaction-diffusion-advection equation to model the dynamical behavior of a single species under water action where d is the diffusion rate and a is the water speed, and they obtained the follow-ing explanation for the above paradox:sufficient random movement can balance the directed movement caused by water flow and lead to population persistence. Later on, Vasilyeva, Lutscher [149] studied the same equation but with Neumann bound-ary condition at downstream end (i.e., ux(L,t)= 0) and obtained similar results. Recently, Lutscher, Lewis, McCauley [111] deduced a more general downstream end boundary condition, dux(L,t) - αu(L,t)= bαu(L,t), where b denotes a loss rate relative to flow rate. Different values of b reflect different environments, for ex-ample,b= 0 means no loss and corresponds to the no-flux condition; b= 1 means the complete loss of water flow and corresponds to free flow condition (Neumman condition); b â†'>∞ means severs loss and corresponds to hostile condition. The first part of chapter 2 of this paper studied the above single equation with such general boundary condition. We derive sufficient and necessary conditions for the persis-tence of a single species in terms of critical habitat size and critical advection rate, and we find a transition phenomenon occurs at b=1/2. To pursue further the mech-anism behind the evolution of dispersal rates, the second part of this chapter then considered a corresponding two-species competition model under the assumption that the two species are identical except their diffusion rates. By spectral theory and some mathematical skills, we obtain complete understanding for 0<≤ b≤ 1, which suggests that slower diffuser will be displaced and larger diffusion rate is selected, extending an earlier work by Lou, Lutscher [105]. While for b> 1, the dynamics becomes quite different, and in particular, we illustrate that for b>3/2, some intermediate diffusion rate may be favored. We mention here that due to the introduction of the parameter b in the boundary condition, the local stability of the semi-trivial steady states cannot be established by the arguments used in [105], but the new method developed in this chapter can be equally applied to deal with the case b=1. Moreover, to obtain the global dynamics of the system for 0≤ 6≤ 1, it seems non-trivial to prove the non-existence result of any co-existence steady state, for which we introduce some new ideas and techniques. These results have been published, see paper 1 in Appendix two.Genetic evolution is a central topic in population genetics. Nonlinear reaction-diffusion equations have arisen numerous applications in population genetics since Fisher 1937 ([56]). One such model dealing with two types of genes is as follows ([125]) where dâ–³u and g(x)u(1 - u)(1+h - 2hu) denote the migration and selection effects respectively, and h specifies the degree of dominance:|h|< 1, no dominance; |h|= 1, complete dominance;|h|> 1, overdominance. In early times, Fleming [57] (1975) and Senn [138] (1983) studied the case|h|< 1 by function analysis and bifurcation theory, respectively, and obtained the existence and stability of nontrivial steady states. More recently, Ni, et. al [127,110] systematically studied the degenerate case h=- 1, and obtained more detailed results on the existence, stability, and limiting profiles of nontrivial steady states (actually their method works for|h|≤ 1). Motivated by these work, chapter 3 of this paper studied the case|h|> 1, for which there appears a third trivial solution u=1h/(2h) (0,1).Beside some parallel results to that in [127,110], we also observe certain new phenomena, for instance, by a specific example we illustrate that when the diffusion rate is small, this problem admits a set of solutions that oscillate around u=(1+h)/(2h)and thus have no limiting profile. Moreover, by spectral analysis, bifurcation theory and upper and lower solution method, we get a better understanding on the dynamics, which shows that the sign of ∫Ω g(x)dx and the size of d play an important role in determining the distribution of gene frequency. We also find that our main results are consistent with that in [127,110] by letting h â†'> - 1. These results have been published, see paper 3 in Appendix two.Biological invasion is an important problem in population ecology. In 2010, Du et. al [44,40] proposed and studied the following free boundary problem, which describes the invasion process of a new or invasive species with the free boundary representing the expanding front [44] considered the homogeneous environment, i.e., m(x)= m> 0, proved that the species either establishes itself successfully(h(t) â†' ∞o and â†' m0, called "spreading"), or fails to establish(h(t â†' h∞<∞ and uâ†' 0, called "vanish-μ ing"), derived a sharp criteria for spreading and vanishing by using h0 and and obtained the asymptotic speed of the free boundary k0= limâ†'∞h(t)/t.Later on, [40] extended these results to higher dimensional and radially symmetric case in weak heterogeneous environment, i.e.,0< m1<m(r)< m2, r=|x|, x ∈Rn. It is worth further considering a more realistic situation where the environment contains both favorable regions{x:m(x)> 0} and unfavorable regions{x:m(x)< 0} ([20]). With this in mind, chapter 4 of this paper studied the more complex case of strong environmental heterogeneity, i.e., m(x) may change sign, for which it seems very hard to investigate the dependence of the principal eigenvalue on the parameter ho, and so some arguments developed in [44,40] no longer work. By regarding D as a variable parameter, we first figure out the dependence of the principal eigenvalue on D, and then derive sufficient conditions for both species spreading and vanishing by upper and lower solution method. This idea works also for the weak case, and we obtain a sharp criteria for species spreading and vanishing (which is parallel to that in [40]). Moreover, under more general conditions, we obtain the asymptotic spreading speed of the free boundary, extending the previous work. Our results show that small diffusion rate is always beneficial for species spreading, while for large diffusion rate, species spreading or not will depend on the initial size u0 and the parameter μ. These results have been published, see paper 2 in Appendix two.Blowup phenomenon is an important research object in combustion theory. In 1970s and 1980s, blowup theory of nonlinear evolution equation received rapid development, and one typical combustion equation of parabolic type is as follows It has been proved that (see [61,73,86,155]) for p≤ 1+2/n, all nontrivial solutions of the above equation will blow up in a finite time, while for p> 1+2/n, there may appear global solutions provided the initial data is very small, where 1+2/n is the Fujita’s critical exponent. In 2001, French mathematician Souplet and his collaborators [55,63] studied the above equation in a moving domain They presented the blowup criteria by energy method, and firstly obtained the global fast solution(h(t) â†'> h∞<∞ and u â†' 0) and global slow solution(h(t) â†' ∞ and u â†' 0) for any p> 1, which greatly enrich previous results obtained in fixed domain problems. Inspired by [55,63], chapter 5 of this paper tries to extend these results to the following combustion model with nonlocal reaction term and double free boundary fronts (Actually, nonlocal in space is only one type of nonlocal terms in combustion theory, see [143] for more details.) Nonlocal reaction term causes certain difficulty:on one hand, it makes the previous energy method loss efficiency, and we need a different way to give the blowup condition; on the other hand, it makes the a priori estimates more complex due to the involvement of the free boundaries. Moreover, double fronts cause more straightforward mathematical difficulty since it is not clear whether the two fronts will converge to finite limits or go to infinity at the same time. We will use some analytical skills to overcome these difficulties, and our results indicate that large initial data guarantees blowup, while the global fast solution exists for sufficiently small initial data, and the intermediate case with a suitably large initial data gives the existence of the global slow solution. These results have been published, see paper 5 in Appendix two.