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

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