The Critical Exponent And Life Span Of Solutions To The Nonlinear Parabolic Equation

The asymptotic behavior of solutions for two types of nonlinear parabolic equa-tions was studied in this paper,it mainly applied the critical Fujita exponent and the second critical exponent to describe the existence and the non-existence of the solutions for equations.Moreover,the life span of non-global solution was also considered.The critical Fujita exponent is used to distinguish global and non-global solutions,and the second critical exponent is used to distinguish global and non-global solutions in the region where they coexist by determining the decay order of the initial value.Further-more,life span refers to the maximum existence interval of the solution by estimating the blow up time of the non-global solution.Firstly,the Cauchy problem of quasilinear parabolic equation with nonlocal weight-ed source ut=?um+(?RnK(x)uq(x,t)dx)p-1/qur-1 was studied.The critical Fu-jita exponent pc=m+2q(1 m)(n s)nqr/nq-(a-s)-and the second critical exponent a*=2q+(p-1)(n-s)-/q(p-r-m)were obtained by applying the Kaplan method and the upper and low-er solution method.It was also shown that the smaller the kernel function K(x),the more conducive to the global existence of the solution and the larger the parameter q,the more conducive to the global existence.Furthermore,this phenomenon was shown by some numerical examples.Secondly,the Cauchy problem of semilinear heat equation with degenerate func-tion ut-div(w(x)?u)= up was considered.Recently,Fujishima et al.Gave the critical Fujita exponent pc=1+2-?/n(??{b1,b1).We further studied its second critical exponent a*=2-?/p 1 to improve the conclusion of the critical exponent for this equation.At the same time,the life span of solution and the influence of the degener-ated coefficient w(x)on solutions are also discussed one by one.The corresponding conclusion was verified finally by several numerical simulation images.