A Theorem of La Salle–Lyapunov Type for Parabolic Systems

This paper deals with the boundary value problem for a nonlinear system of parabolic differential equations for $u = u(t,x)$\[ u_t = Lu + f(u)\quad {\text{in }}\Omega ,\quad u(0,x){\text{ given}},\quad \frac{{\partial u}}{{\partial \nu}} = 0\quad {\text{on }}\partial \Omega \] under the assumption that a Lyapunov function $V(z)$ for the corresponding ordinary differential equation system $u' = f(t,u)$ exists. In the case where L is one and the same selfadjoint elliptic operator of second order for all components of u, the real-valued function $U(t,x) = V(u(t,u))$ satisfies a parabolic differential inequality \[ U_t \leqq LU - c\left| {u_x } \right|^2 \qquad (c > 0).\] It follows that u exists globally and is bounded if $u(0,x)$ is bounded. The limit set $\Lambda ^ + $ (as $t \to \infty $) of any solution u is nonempty and compact, it consists of constant functions only, it is an invariant set for $u' = f(u)$, and $\dot V = V_z \cdot f$ vanishes on $\Lambda ^ + $ (analogue of La Salle’s stability theorem for ordinary differential equations). The results are then extended to quasilinear systems where $Lu = (a_{ij} (x,u)u_{x_j} )_{x_j} $. In the case where different elliptic operators are involved, $u_t^k = L^k u^k + f^k (u)(k = 1, \cdots ,n)$, it is assumed that $a_{ij}^k (x) = c^k (x)a_{ij} (x)$ with $c^k > 0$. A Lyapunov functional $U(t) = \int_\Omega V (u(t,x)) dx$ is employed, but the boundedness of solutions has to be assumed or obtained by other means.