Abstract The purpose of this paper is to prove the global existence in time of solutions for the coupled reaction-diffusion system: { ∂ u ∂ t − a Δ u − b Δ v = f ( u , v ) in ] 0 , + ∞ [ × Ω ∂ u ∂ t − c Δ v = g ( u , v ) in ] 0 , + ∞ [ × Ω * * 487 * * $$\left\{ {\matrix{{{{\partial u} \over {\partial t}} - a\Delta u - b\Delta v = f\,(u,\,v)} \hfill & {{\rm in}\,]0,\, + \infty [ \times \Omega } \hfill \cr {{{\partial u} \over {\partial t}} - c\Delta v = g(u,v)} \hfill & {{\rm in}\,]0,\, + \infty [ \times \Omega } \hfill } } \right.$$ with triangular matrix of diffusion coefficients. By combining the Lyapunov functional method with the regularizing effect, we show that global solutions exist. Our investigation applied for a wide class of the nonlinear terms f and g.