Bifurcation Theory And Its Applications In Semilinear Partial Differential Equations

In the field of nonlinear partial differential equations, there exists an importantnonlinear phenomenon, which is bifurcation phenomenon. So called bifurcation phe-nomenon means that, when the parameters cross through certain critical values, thephenomenon of the change of some structural properties in the system are studied.Bifurcation is made up of three parts: local bifurcation, semi-local bifurcation andglobal bifurcation. The study of bifurcation of partial differential equation is not onlyrelated to the theories of classical dynamical system, but also related to the otherknowledge such as topology, algebra and functional analysis. The study is of greattheoretical significance and practical background.This paper mainly performs local Hopf bifurcation analysis, global steady statebifurcation analysis, and Turing bifurcation analysis to the semilinear partial differ-ential equations by using center manifold theory, normal form methods and globalsteady state bifurcation theorems. The main contents of the paper are as follows:1,By using the center manifold theory and normal form methods, we obtainan abstract local Hopf bifurcation theorem for the semilinear partial differential equa-tions, which derives the conditions for the general semilinear reaction-diffusion equa-tions to undergo Hopf bifurcation, and obtains the general algorithm to determinethe bifurcation direction and the stability of the bifurcating periodic solutions. Thismakes it convenient for readers'future applications in the Hopf bifurcation analysisin semilinear partial differential equations. By applying our results to the diffusivepredator-prey system and the diffusive Lengyel-Epstein system modeling the CIMAchemical reaction, we prove that these two semilinear partial differential equationsnot only have the spatial homogenous periodic solutions but also have the spatial non-homogenous periodic solutions. Moreover, the"size"of the spatial domain affects thenumber of the spatial non-homogenous bifurcating periodic solutions in the followingway: whenever the domain is large enough, the larger the domain is, the more thespatial non-homogenous bifurcating periodic solutions.2,By using Shi and Wang's generalized global steady state bifurcation theorem,which says that under suitable conditions the local steady state bifurcation is equiv- alent to the global steady state bifurcation, we derive an abstract simplified globalsteady state bifurcation theorem for the semilinear partial differential equations. Byapplying our results to a diffusive homogenous predator-prey system, we prove that,under certain conditions, this system has a global steady state bifurcation, and partic-ularly we show that, under suitable conditions, the system may have loops consistingof the positive steady solutions, which connects with two different bifurcation points.Further, we consider the interaction between Hopf bifurcations and steady state bifur-cations.3,By using the general results on Turing bifurcations of the general semilinearpartial differential equations, we consider the Turing bifurcation of a Lengyel-Epsteinsystem in CIMA chemical reaction, in a bounded interval in the one dimensionalspace, and obtain the criterion for the system to occur Turing bifurcation. Finally wepresent numerical simulations to support our theoretical analysis.