Multi-valued random dynamics of partly dissipative reaction–diffusion system with discontinuous nonlinearity on RN

This paper is devoted to studying a system consisting of a reaction–diffusion equation with multi-valued right-hand side and an ordinary differential equation in absence of dissipation term, which is defined on the whole space R N . The system is driven by time-dependent forces and coloured noise with nonlinear diffusion. We first establish the global existence of strong/mild solutions for initial-value problems. The measurability of solution map with respect to sample points and initial values is then obtained via the upper semicontinuity, which indicates that these solutions define a (non-autonomous) multi-valued random dynamical system. Finally, we prove the existence of pullback attractor for the dynamical system.