Fourier Analysis of a Dissipative Plate Equation with Rotational Inertia: A Mathematical Framework for Biological Modeling.

<jats:p>This paper studies the linear Cauchy problem for a dissipative plate equation with rotational inertia in Rn. The model combines fourth-order bending effects, frictional damping, and rotational inertia, and is relevant as a linear analytical component for dissipative processes occurring in thin biological structures such as soft tissues, cell membranes, epithelial layers, and bio-inspired elastic interfaces. The analysis is developed in a precise Sobolev-space framework. After defining the functional setting and the solution class, the partial differential equation is transformed into a family of frequency-dependent second-order ordinary differential equations. The characteristic roots are computed explicitly, the fundamental Fourier multipliers are derived, and their low-frequency asymptotics are established. The results show that the solution separates into a slow quartic-diffusive mode and a faster dissipative mode. The paper also clarifies its relationship with existing Fourier-based studies on dissipative wave and plate equations, especially works on fractional damping and plate equations with rotational inertia. The contribution is a self-contained, biologically motivated, and mathematically rigorous formulation of the pure rotational-inertia plate model, with explicit solution operators that can support future nonlinear and applied biological modelling.</jats:p>