On intermittency of stochastic heat equation withDirichlet boundary condition

来源 :2016随机微分方程和随机过程研讨会(Workshop on SDEs and Stochastic Processes | 被引量 : 0次 | 上传用户:listen342325
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  In this talk,unper some proper conditions,I will consider some propertiesabout intermittency of the following stochastic heat equation(e)t u(t,x)=1/2△u(t,x)+λσ(u(t,x))(w)(t,x)with homogeneous Dirichlet boundary condition on [0,1], wherew(t,x)denotes the space-time white noise.I am mainly interested in some properitiesrelative to the noise intensityλ>0,which arepeculiar to Dirichlet boundarycondition.One is about stability and unstability of the solution for fixedλ.It is proved that for each small enough λ,the p-th moment ofsupx∈[0,1]|u(t,x)| is exponentially stable,however,for each large enoughλ,p-th moment of supx∈[0,1]|u(t,x)| is unstable and grows at an exponential rate.The other is to study the excitability of the noise as λ→∞ for each t>0.Based on the global estimate of the heat kernel,it is proved that the noise excitation of the p-th energy of u(t,x)is equal to 4.
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