Complicated Asymptotic Behavior Of Solutions For Porous Medium Equations With Absorptions Or Sources

In this monograph, we consider the Cauchy problem of the porous medium equation (PME) with absorptions or sources where m>1,p>m+2/N.,αare constants. Our main purpose is to study the asymptotic behavior of solutions, especially the complicated asymptotic behavior. This monograph is divided into five chapters. The first chapter is the introduction of this monograph. We will s-tudy the porous medium equation without absorptions and sources, with absorptions and with sources in the second chpapter, the third chapter and the fourth chapter respectively. The conclusions will be summarized in the last chapter.Complicated asymptotic behavior of solutions for evolutionary equations is a new hot topic of mathematicians in recent years. It were, in 2002, Vazquez and Zuazua who first found this phenomenon in an article in commemorating J. L. Lions. For the Cauchy problem of the heat equation, from 2003 to 2007, by using the linearity and the explicit expression of solutions, Cazenave, Dickstein and Weissler studied the complicated asymptotic behavior of the rescaled solutions tμ/2u{tβ.,t) (μ,β>0) with gobal uniform convergence. For the Cauchy problem of the porous medium equation, we get the propagation ve-locity estimates on solution support with initial value u0∈L1(RN) or u0∈L∞(RN) in the second chapter first and then use these propaga-tion velocity estimates, the relations between the PME semigroup op-erator and the scaling operator to investigate the complicated asymp-totic behavior of the rescaled solutions tμ/2u(tβ., t) (μ,β>0) with gobal uniform convergence.In 2002, Vazquez and Zuazua utilized the scale invariance, the regularity and the continuity of semigroup generated by the Cauchy problem of the heat equation, the porous medium equation and other evolution equations to study the complicated asymptotic behavior of solutions, and they revealed that the asymptotic behavior of these so-lutions in Lloc∞(RN) and the spatial asymptotic behavior of initial value in L∞(RN) have some equivalent relations. Later, Cazenave, Dickstein and Weissler found that for the heat equation, the asymptotic behavior of solutions in L∞(RN) and the spatial asymptotic behavior of initial value in Wσ(RN) also have similar equivalent relations as the above. In the third chapter, we find that for the porous medium equation with absorptions or without absorptions and sources, the asymptotic behavior of solutions in L∞(RN) and the spatial asymptotic behavior of initial value in Wσ(RN) also possess the similar equivalent relations as the heat euation. For these purposes, we need to obtain the de-cay estimates on solutions and the propagation velocity estimates on solution support for the porous medium equation with initial value n0∈Wσ(RN). As an important application, we find that the resluts, which had already been established by Alikakos, Rostamian, Kamin and Peletier in eighties and nineties of the last century, that the so-lution of the porous medium equation with absorptions (or without absorptions and sources) uniformly converge to a function in the cone {x∈RN;|x|< Ct(?)/σ(m-1)+2} is also true if the cone is replaced by the whole space.In the fourth chapter, we first get the global solutions and the decay estimates on solutions about the porous medium equation with sources and then study the complicated asymptotic behavior of these solutions. As a direct application, we give a simple proof of the result-s obtained by Mukai and Mochizuki in the beginning of this century that if initial value u0(x)∈Bη(M)σ,+, then the source term is negligible in the asymptotic behavior of solutions.