一类抛物型方程的熄灭

Extinction of the Solution on a Type of Parabolic Equation

解的熄灭现象是非线性抛物型方程解的一个重要性质,有着广泛的物理背景。受文[1]启发,在文[3]的基础上,采用能量估计的方法,讨论了一类抛物型方程初边值问题ut+(-Δ)2u+λ|u|γ-1u-βup=0,(x,t)∈Ω×(0,∞)′uvi|Ω×(0,∞)=0,i=0,1u(x,0)=u0(x),x∈Ω解的渐近性态。得到当0<γ<p<1对任意u0(x)∈W20,2(Ω)∩Lp+1(Ω),存在λ0,λ>λ0时,以上方程的解在有限时间熄灭。在此基础上,本文还给出了解的能量估计。

The extinction phenomenon of solution is one of the important qualities of nonlinear parabolic partial differential equation, as it explains comprehensive realistic backgrounds. Enlightened by paper [ 1 ] and after the work of paper [ 3 ], extinction of the solution of initial - boundary value problem for the parabolic equation {偏du/{偏dt=（-△）^2u＋λ｜u｜^γ-1u-βu^p=0,（x,t）∈Ω×（0,∞） 偏d′u/偏dv^i｜偏dΩ×（0,∞）=0,i=0,1 u（x,0）=u0（x）,x∈Ω is studied. It is found that when 0 〈 γ 〈 p 〈 1 , for all u0（x）∈W0^2,2（Ω）∩L^p＋1（Ω） there exists λ0, when 〉 λ0 the solution for the above equations extincts in finite time. Further more, the energy estimate for the above equations are given out.