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We study a class of nonlinear nonlocal cochlear models of the transmissionline type, describing the motion of basilar membrane (BM) in the cochlea. Theyare damped dispersive partial differential equations (PDEs) driven by timedependent boundary forcing due to the input sounds. The global well-posednessin time follows from energy estimates. Uniform bounds of solutions hold in caseof bounded nonlinear damping. When the input sounds are multi-frequency tones,and the nonlinearity in the PDEs is cubic, we construct smooth quasi-periodicsolutions (multi-tone solutions) in the weakly nonlinear regime, where newfrequencies are generated due to nonlinear interaction. When the input is twotones at frequencies $f_1$, $f_2$ ($f_1 < f_2$), and high enough intensities,numerical results illustrate the formation of combination tones at $2 f_1 -f_2$and $2f_2 -f_1$, in agreement with hearing experiments. We visualize thefrequency content of solutions through the FFT power spectral density ofdisplacement at selected spatial locations on BM.

Global well-posedness and multi-tone solutions of a class of nonlinear nonlocal cochlear models in hearing