  A new mathematical model of memristive neural networks described by the
partly diffusive reaction-diffusion equations with weak synaptic coupling is
proposed and investigated. Under rather general conditions it is proved that
there exists an absorbing set showing the dissipative dynamics of the solution
semiflow in the energy space and multiple ultimate bounds. Through uniform
estimates and maneuver of integral inequalities and sharp interpolation
inequalities on the interneuron differencing equations, it is rigorously proved
that exponential synchronization of the neural network solutions at a uniform
convergence rate occurs if the coupling strength satisfies a threshold
condition expressed by the system parameters. Applications with numerical
simulation to the memristive diffusive Hindmarsh-Rose neural networks and
FitzHugh-Nagumo neural networks are also shown.
