Critical Exponents For A Class Of Fast Diffusion Equations With Coupled Boundary Sources

This paper is concerned with the nonlinear coupled boundary value problem for a class of fast diffusive system.We focus our attention to the discussions of the critical exponents theory,and try to give a complete characterization for these exponentsα,β,p,q.Comparing with former works for the fast diffusive system,the problem we considered not only include the coupled parameters,but also include the non-coupled parameters,which makes the study more challenging and involved to a certain extent.As is well known,Newton Filtration system and Non-Newton Filtration system are two typical cases of the polypropic filtration system,and described different filtration processes,which both have something in common and some difference not only in the Study of Manner,but also the results.For instance, they both have singularity and degeneracy,but the reason caused that is different; and they both have no classical solution,so the weak solutions are considered,however,the definition and regularity are different for the two kinds of systems.In the first chapter and second chapter,we shall consider Newton Filtration system and Non-Newton Filtration system respectively.As two important classes of parabolic systems,which appear in many fields.So, to carry out research on the two kinds of systems have important academic significance and application value.We shall emphasize particularly on the approach and technic,and try to supply some thoughts for some general systems. In this paper,we mainly by using upper and lower solutions approach, and combining with comparison(see for example[11]-[15]),to discuss the blow-up property of solutions.This approach is often to be used to study the critical Fujita exponent,and the key is to construct suitable upper and lower solutions.In this paper,we mainly by construct a series of self-similar upper and lower weak solutions to study the large time behavior of solutions.Due to the nonlinearity caused both by diffusion and by the coupled boundary condition,which makes the study more difficulty and challenging.In the third chapter,by using the approach employed in Chapterâ… and chapterâ…¡,we shall consider more general case,that is the polypropic filtration system with coupled nonlinear boundary conditions,namely the case 0＜m₁(p₁-1)＜1,0＜m₂(p₂-1)＜1,α＞0,β＞0.We shall character the critical exponents and critical curves by the coupled parameters and noncoupled parameters precisely.We shall see that the critical exponents for the non-coupled parameters areα_c=(m₁+1)(p₁-1)/P₁,β_c=(m₂+1)(p₂-1)/p₂, which have the following properties:(1) Ifα＞α_c orβ＞β_c,then the solution blow up for any p,q＞0;(2) If 0＜α＜α_c and 0＜β≤β_c,then there exists p,q＞0 such that all nonnegative solutions will exist globally.For Case(2),we need to make further discussion.We aimed to study the global existence and the blow up properties of solutions under the assumption that 0＜α＜α_c,0＜β≤β_c.Precisely speaking,we showed that if 0＜α≤α_c and 0＜β≤β_c,the critical curve of global existence is and the Fujita curve is whereSumming up,we discussed the critical exponents characterizing the blowup and global existence properties for a class of fast diffusive system with nonlinear coupled boundary sources,and obtain the critical exponents or critical curve given by the non-coupled parameters and the coupled parameters. Comparing with the already known works,the problem we considered not only include the coupled parameters,but also include the non-coupled parameters, the appearance of which bring more challenges and make the problem more complicated.Especially,we need to search the critical exponent for the noncoupled parameters,which is the essential differences between the problems we considered and the known works.