Well-posed ness of non-linear systems when coupled with diffusive systems

A family of models has been proposed both for visco-elasticity or elastoplasticity, mostly in finite dimension, damped by a general class of therrnodynamically consistent models: they involve causal pseudo-differential time-operators of diffusive type - such as fractional integrals or derivatives - which can be equivalently reformulated as controlled-and-observed heat-like equations in an unbounded domain. Hence, such systems are both infinite-dimensional and nonlinear, The main difficulty lies in the non compactness of the diffusive part: we present new existence and uniqueness results on such pseudo-differentially damped ODEs and PDEs, together with numerical simulations.