We prove the existence and boundedness of global classical solutions for the semilinear heat equation \(u_t-\Delta u=V(x)|u|^{p-1}u \) in \(\Omega \) , where \(p>\frac{n+2}{n-2}\) , \(\Omega \subset \mathbb {R}^n\) is a bounded convex domain, \(n\ge 3\) . If \(V(x)\in C^1(\overline{\Omega })\) and the initial date satisfies some decay properties, then we can show that any global classical solution has to decay in time faster than \(t^{-\frac{1}{p-1}}\) . The method we use is based on weighted energy estimates of Giga-Kohn and some properties of Morrey spaces and the quasi-monotonicity formula of weighted energy.
