Global existence and asymptotic decay for small solutions of general quasilinear hyperbolic balance laws

This paper establishes global existence and asymptotic decay for small solutions to quasilinear systems of hyperbolic balance laws, where, generalizing previous works, the hyperbolic operator does not need to admit an entropy nor does the source term need to satisfy any symmetry assumptions. Dissipative properties are characterized by three conditions corresponding to regimes of small, intermediate and large wave numbers in Fourier space, and the fully nonlinear system is treated by using methods of para-differential calculus recently developed in the context for proofs of global existence and decay in second-order hyperbolic systems. This work leads, in particular, to asymptotic stability of rest-states for multidimensional Jin–Xin relaxation system, a result not accessible through previous methods.