Abstract: In this paper, we are concerned with the Cauchy problem:
u_t=△u+|u|^γ-1u-u x∈Rⁿ,γ>1,t∈R⁺ u(x,0)=u₀(x) in D_q^α. We got some existence and asymptotic results of their global solutions. Additionally, we got the global existence theorem of the Cauchy problem
u_t=△u+|u|^γ-1u x∈Rⁿ,γ>1,t∈R⁺ u(x,0)=u₀(x) in D_q^α, where q∈[(1)/2n(γ-1),∞),α=(2)/γ-1-(n)/q, in the case(1)/2n(γ-1)>1.