Asymptotic Behavior Of Singular Solutions To Heat Systems With Nonlinear Absorptions

This thesis deals with asymptotic behavior of singular solutions for heat systems with nonlinear absorptions.There are two topics included:Firstly,quenching behavior of solutions for heat systems coupled via singular absorptions;Secondary,multiple blow-up rates for multi-coupled heat systems with absorptions.The thesis composes of four chapters:In Chapter 1 we summarize the background of the related issues and state the main results of the present thesis.Chapter 2 deals with finite time quenching for one-dimensional heat systems coupled via singular boundary absorptions of the form u^-p1v^-q1 and u^-p2v^-q2.Besides the coupling factors v^-q1 and u^-p2,there are two additional singular boundary absorption factors, namely,u^-p1 and v^-q2 for the components u and v respectively.We at first propose a criterion to distinguish simultaneous and non-simultaneous quenching,and then determine all possible quenching rates in the model,three for simultaneous quenching and two for non-simultaneous quenching,under different nominations.This shows real contributions of the two additional singular factors to the quenching behavior of solutions.Chapter 3 mainly discuss quenching behaviors of solutions to heat equations with coupling singular absorptions u^-m + v^-p and u^-q + v^-n.We study them in two cases respectively, inner absorption couplings and boundary absorption couplings.For both cases, we at first identify simultaneous and non-simultaneous quenching,and then establish four kinds of simultaneous quenching rates,which are uniformly represented via the characteristic algebraic system introduced for the model.It is interesting to find that there are two simultaneous quenching rates which share a common subregion of parameters,determined by the initial data assumed.Chapter 4 studies finite time blow-up problem to a multi-coupled heat system with absorptions.In the first part we consider the case of balanced absorptions,and obtain four absorption-independent simultaneous blow-up rates under different dominations of nonlinearities.Two characteristic algebraic systems associated to the problem are introduced to get very simple descriptions for the four simultaneous blow-up rates as well as the critical blow-up criterion,respectively.The second part deals with unbalanced absorption case(i.e.,stronger absorption for one component and weaker absorption for the other).We prove that if the positive coupling sources dominate the system with unbalanced absorptions,then there are also eight possible absorption-related blow-up rates for the model,besides the four absorption-independent ones shown above.This is substantially different from the scalar equations with absorptions,whose blow-up rates are all known as absorption-independent.