Study of global solutions of parabolic equations via a priori estimates. Part I: Equations with principal elliptic part equal to the laplacian

We study global solutions of u_t=Δu + f(u, Du) with zero boundary data. Under suitable hypotheses on f, we show that an energy, which is roughly the H¹ norm of u, cannot change too rapidly. When f depends only on u, this estimate implies that all global solutions are bounded.