We are concerned with the existence of nonnegative periodic solutions of \(u_t = \Delta u^m + h(t) f(u)\) in \(\Omega \times \mathbb{R}\) with the Dirichlet boundary condition, where \(m > 1\), \(\Omega\) is a smoothly bounded domain in \(\mathbb{R}^N\) and \(h\) is a given positive periodic function on \(\mathbb{R}\). The forcing term in the paper satisfies \[ \liminf_{\rho \to \infty} h(t) {f(\rho) \over \rho^m} > \lambda_1 \quad \text{uniformly in } t \in \mathbb{R}, \] for example, \(f(u) = u^p\) with \(p \geq m\) and \(h > \lambda_1\). We adapt the Leray-Schauder degree theory in the proof. To do that, we need an a priori estimate for solutions in \(L^\infty\). Inequalities of Harnack type and the blow up (or scaling) argument play crucial parts.