Propagation Dynamics Of Degenerate Reaction-diffusion Equations

Since the Allee effect is inevitable,much attention has been paid to the corre-sponding reaction-diffusion equations,in which the propagation dynamics has been widely studied.Comparing with the nondegenerate reaction-diffusion equations,it has been proved that there are some essential differences on the traveling wave solu-tions and spreading speed in degenerate case,which include the uniqueness of wave speed and asymptotic behavior for the traveling wave solutions,the success or failure of asymptotic spreading.Different from the nondegenerate case,a zero eigenvalue often arises in the eigenvalue problem of the corresponding linear operator,this leads to the difficulties in applying the classical results directly in degenerate case.The purpose of this thesis is to investigate the traveling wave solutions and spreading speed of degenerate reaction-diffusion equations with time periodic and delay as well as a Lotka-Volterra type competition system with degenerate nonlinearity.We first study the traveling wave solutions and asymptotic spreading of a class of time periodic diffusion equations with degenerate nonlinearity.The asymptotic behavior of traveling wave solutions is established by using auxiliary equations and a limit process.Different from the nondegenerate case,our results reflect that the traveling wave solutions with critical speed decay exponentially,while those with non-critical speed do not decay exponentially.With the help of asymptotic behavior,the monotonicity and uniqueness,up to translation,of traveling wave solutions with critical speed are determined.Combining super and sub-solutions and the stability of steady states,some sufficient conditions on asymptotic spreading are also given,which reveal that the success or failure of asymptotic spreading depends on the degeneracy of nonlinearity as well as the size of compact support of initial value.We then study a degenerate diffusion equation with time delay.In the de-generate monostable case,the existence of traveling wave solutions is proved by constructing proper super-solutions,and the monotonicity,asymptotic behavior as well as the uniqueness up to translation of traveling wave solutions are also estab-lished.Then the role of delay and degeneracy on the critical speed of traveling wave solutions and success or failure of asymptotic spreading are explored.Our results reveal that the success or failure of asymptotic spreading depends on the degen-eracy of nonlinearity as well as the size of initial value,this is different with the nondegenerate case.To illustrate our results,the traveling wave solutions of several degenerate equations are investigated,which show that the large delay could slow down the critical speed.Lastly,we consider the asymptotic spreading of a Lotka-Volterra type com-petition diffusion system in degenerate case.For the classical monostable case,extinction or persistence phenomenon is completely determined by the dynamics of the corresponding kinetic system,while the size of initial habitat does not af-fect the final states.Under the degenerate case,we present various extinction or persistence results by selecting different initial values although the corresponding kinetic system is fixed,which also imply the balance between degenerate nonlinear reaction and diffusion.For example,even if the positive steady state of the corre-sponding kinetic system is globally asymptotically stable,we observe four different spreading-vanishing phenomena by selecting different initial values.In addition,the interspecific competition may be harmful to the persistence of one species by taking proper initial values.and the superior competitor in the sense of the corresponding kinetic system is not always unbeatable,it can be wiped out by the inferior com-petitor in the sense of the corresponding kinetic system,which depends on the size of initial habitat as well as the intensity of Allee effect.