Boundedness of the solutions of a nonlinear elliptic system in a cylindrical domain (Q1274119)

The author studies the boundary value problem \[ a(\partial^2_t u+\Delta u) +\gamma \partial_t u-f(u,t)= g(t), \quad u|_{t=0}=u_0,\;u |_{\partial w} =0 \] in the half-cylinder \([0,\infty) \times w\), \(w \subset \mathbb{R}^n\); \(u(t,x)\in\mathbb{R}^k\). He proves that all solutions of this problem are bounded and presents an a priori estimate.