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Section 3 is addressed to proving an interpolation inequality by virtue of Carleman estimates for the parabolic equations. 2, we give the main results in this paper. The rest of this paper is organized as follows.

Since there is no elliptic-type equation in our situation, the interpolation inequality we obtain here differs from. Where \(c(\cdot )\in L^+\Delta )\)”, therefore, we have to get an interpolation inequality for a system coupled by four parabolic-type equations (see ( 4.5)).