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Ariel So
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stability of parabolic problems where nonhomogeneous term in $L^2(0,T; H^{-1}(\Omega))$
@Jean Duchon Thanks for your suggestion. It really needs $f\in L^2（0，T; L^2(\Omega))$ to complete the proof of stability result. The assumption $f\in L^2(0,T; H^{-1}(\Omega))$ is not enough.
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stability of parabolic problems where nonhomogeneous term in $L^2(0,T; H^{-1}(\Omega))$
I am following a proof where $f$ is assumed to be in the space $L^2(0,T; L^2(\Omega))$ , I have presented the proof in the following link. dropbox.com/s/n5j0k6kr3rmgq2a/Proof%20.pdf?dl=0
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stability of parabolic problems where nonhomogeneous term in $L^2(0,T; H^{-1}(\Omega))$
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stability of parabolic problems where nonhomogeneous term in $L^2(0,T; H^{-1}(\Omega))$