Abstract: The methods of L^p estimation are used to discuss the extinction phenomena of the solutions to the following reaction-diffusion equations with initial-boudnary values ∂u/∂t=Δu-λ|u|^γ-1u-βu ((x,t)∈Ω×(0,+∞)), u(x,t)|_{∂Ω×(0,+∞)}=0, u(x,0)=u₀(x)∈H₀¹(Ω)∩L^1+γ(Ω) (x∈Ω).Sufficient and necessary conditions about the extinction of the solutions is given. Here λ>0, γ>0, β>0 are constants, Ω∈R^N is bounded with smooth boundary ∂Ω. At last, it is simulated with a higher order equation by using the present methods.

Key words: reaction-diffusion equation, extinction, L^p estimation, Bernoulli equation