Asymptotic Behaviors Of Solutions To Cauchy Problem Of Coupled Convection-diffusion Equations

This paper concerms the asymptotic behavior of solutions to the Cauchy problem of a.class of coupled semilinear parabolic systems with gradient terms.Using the energy comparison method and principle,the blow-up theorem of Fujita type is established and the critical Fujita curve is formulated by spacial dimension,the behavior of the coefficient of the gradient term at infinity.In the first part we study the coupled parabolic systems of second order with gradient terms.The systems could describe several diffusion phenomena coming from nature.Firstly we set the definitions of the solutions to the equations.and then we proof the key lemma in the further work.With the self-similinear structure.critical Fujita curve can be proved by the method of super and subsolution.Fujita expo-nents and theorems are established in the last two parts.The blow-wp theorems is shown by energy estimates and the existence of nontrivial global solutions is shown by constructing self-similar supersolutions.In the second part,we study the equation systems with gradient terms which are coupled by reactions.The coefficients of reactions terms have the same limit similarly.We set the definitions of the solutions to the equation systems.we construct self-similar supersolutions.The two new cofficients make the problem more comlplicated.We established the critical Fujita curve on the base of the previous work.In the third part,we study the coupled equation systems with gradient terms which the coefficients of reactions terms have different limits.In order to make the energy estimates for the two solutions having the same grade we should choose different weights for different convections.We prove the blow-up theorem by energy estimates and we prove the existence of nontrivial global solutions by constructing self-similar supersolutions.In the last part,we study the systems of Cauchy problem with gradient terms which are coupled by reactions in a more general condition.It is more complicated for the equation systems we concerned without any self-similar structure.Instead of constructing self-similar subsolutions,we prove the blow-up theorem by energy esti-mating.For the existence of nontrivial global solutions is shown by a series of precise culcalating and construting complex auxiliary supersolutions.We obtain the Fujita exponents and establish the theorems of Fujita type for some coupled parabolic equations with reactions and convections.The results display how the reactions and the convections influence the large time behavior of solutions to the equation systems.