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We consider systems of damped wave equations with a state-dependent dampingcoefficient and perturbed by a Gaussian multiplicative noise. Initially, weinvestigate their well-posedness, under quite general conditions on thefriction. Subsequently, we study the validity of the so-calledSmoluchowski-Kramers diffusion approximation. We show that, under morestringent conditions on the friction, in the small-mass limit the solution ofthe system of stochastic damped wave equations converges to the solution of asystem of stochastic quasi-linear parabolic equations. In this convergence, anadditional drift emerges as a result of the interaction between the noise andthe state-dependent friction. The identification of this limit is achieved byusing a suitable generalization of the classical method of perturbed testfunctions, tailored to the current infinite dimensional setting.

Smoluchowski-Kramers diffusion approximation for systems of stochastic damped wave equations with non-constant friction