For one-dimensional system of dissipative balance laws endowed with a convex entropy we prove, under natural assumptions, that a constant equilibrium state is asymptotically L 2 ?stable under a zero-mass initial disturbance. The technique is based on the construction of an appropriate Liapunov functional involving the entropy and a so-called compensation term.

We consider initial-boundary-value problems for a class of nonlinear third order equations having non-autonomous forcing terms and get new asymptotic stability results by means of the Liapunov second method. The class includes equations arising in Superconductor Theory, Quantum Mechanics and in the Theory of Viscoelastic Materials.