Large time behavior of ODE type solutions to higher-order semilinear parabolic equations with small initial data

Abstract

Let u be a solution to the Cauchy problem for semilinear parabolic equations(Pm){∂tu+(−Δ)mu=|u|αinRN×(0,∞),u(x,0)=λ+φ(x)>0inRN, where m=1,2,⋯,0<α<1,N≥1,λ>0,φ∈BC(RN)∩Lr(RN) with 1≤r<∞. In the case of m=1, by the comparison principle, one can easily show that the solution u to problem (Pm) behaves like a positive solution to ODE ζ′=ζα in (0,∞) as t→∞. In the case of m≥2, because of the lack of the comparison principle, it is not obvious that the solution u to problem (Pm) behaves like a solution to the ODE as t→∞. In this paper, by imposing a smallness condition‖φ‖L∞(RN)<c with a certain constant c dependent on m,α and λ, we prove that the solution u behaves like a solution to the ODE as t→∞. Furthermore, we obtain the precise description of the large time behavior of the solution u and reveal the relationship between the diffusion effect (Pm) has.