The paper is concerned with global existence of solutions to problems involving a semilinear parabolic system \[ v_ t(x,t)=D\Delta v(x,t)+f(v(x,t)),\quad x\in \Omega,\quad t>0, \] where \(\Omega \subset {\mathbb{R}}^ n\), bounded and with a smooth boundary, D is an \(m\times m\) diagonal matrix with positive diagonal entries, \[ v(x,0)=v_ 0(x)\in L^{\infty}(\Omega;{\mathbb{R}}^ m);\quad \alpha_ iv_ i+\beta (\partial v_ i/\partial n)=\gamma_ i,\quad 1\leq i\leq m,\quad x\in \partial \Omega,\quad t>0, \] \(\alpha_ i,\gamma_ i\in {\mathbb{R}}\) and (i) \(\alpha_ k\geq 0\), \(\beta\in \{0,1\}\); (ii) if \(\beta =0\), then \(\alpha_ k=1\), \(1\leq k\leq m\); (iii) if some \(\alpha_ i=0\), then all \(\alpha_ k=0\), \(1\leq k\leq m\), \(\beta =1\), and \(\gamma_ i=0\), \(1\leq i\leq m.\) This problem has local existence and uniqueness and the author has chosen to follow and generalise ideas of \textit{S. L. Hollis}, \textit{R. H. Martin} and \textit{M. Pierre} [ibid. 18, 744-761 (1987; Zbl 0655.35045)] by assuming that the nonlinearity \(f: {\mathbb{R}}^ m\mapsto {\mathbb{R}}^ m,\) a locally Lipschitz function, satisfies a Lyapunov-type condition: There exists \(M\in {\mathbb{R}}\) such that \(\nabla H(Z).f(Z)\leq MH(Z)\), all \(Z\in {\mathbb{R}}^ m\), for H: \(\in^ 3m\mapsto [0,\infty)\), smooth, \(| H(Z)| \to \infty\) as \(| Z| \to \infty.\) This condition gives global existence of solutions to the ODE system \(y'=f(y)\), \(t>0\); \(y(0)=y_ 0.\) The author states conditions on H and f which guarantee global existence for the solution of the parabolic system. The paper also includes applications of this theory to some reaction-diffusion and nerve conduction problems. References include 21 items.