Formulas of Liapunov functions for systems of linear partial differential equations

Formulas of explicit quadratic Liapunov functions for showing asymptotic stability of the system of linear partial differential equations ∂u ∂t = A u(t, x) on (0,∞) × Ω, are constructed, where A is an n × n real matrix, u = (u1,u2, . . . , un)T ,Ω is a bounded domain in Rk with smooth boundary ∂Ω, and denotes the Laplacian operator on Rk with u = ( u1, u2, . . . , un)T . These formulas are also modified and applied to a number of nonautonomous linear and nonlinear systems and models in structural stability, traveling wave, and Navier–Stokes equations. © 2006 Elsevier Inc. All rights reserved