Necessary and sufficient conditions for oscillation of impulsive parabolic differential equations with delays

In this paper, we study a class of impulsive parabolic differential equations with several delays of the form (*){∂u(x,t)∂t=a(t)Δu(x,t)+∑k=1sak(t)Δu(x,t−ρk)−∑i=1mqiu(x,t−σi),t≠tj,u(x,tj+)−u(x,tj−)=bju(x,tj),j∈I∞,(x,t)∈Ω×R+≡G, with the boundary condition (**)∂u(x,t)∂N=0,(x,t)∈∂Ω×R+,t≠tj,j∈I∞. We firstly obtain that every solution of the problem (*) and (**) oscillates in G if and only if every solution of the delay differential equation (***){V′(t)+∑i=1mqiV(t−σi)=0,t≥0,t≠tj,V(tj+)=(1+bj)V(tj),j∈I∞, oscillates. Based on this conclusion, some necessary and sufficient conditions for oscillation of problem (*) and (**) are established.