The author studies the asymptotic behavior of solutions of the Cauchy-Dirichlet problem (with zero boundary data) for the equation \[ u_ t= \Delta u + \lambda u +a(x)u^ p \] with a function \(a\) which is positive on some subdomain of the domain \(\Omega\) on which the problem is presented, and negative on some other subdomain of \(\Omega\); \(p>1\) and \(\lambda\) are real parameters. There are several different situations which arise. First, some special geometry is required for the sets \(\Omega_+\) and \(\Omega_-\) on which \(a\) is respectively positive or negative; each must be the interior of the support of the corresponding positive or negative part of \(a\), and the Dirichlet eigenvalues (for the Laplacian) on \(\Omega\), \(\Omega_+\) and \(\Omega_0=\Omega\setminus (\overline {\Omega_-}\cup \overline {\Omega_+})\), denoted by \(\sigma\), \(\sigma_+\) and \(\sigma_0\), respectively, must be ordered as \(\sigma<\sigma_0<\sigma_+\). (In fact, the first inequality is always true, while the second is true if, for example the measure of \(\Omega_+\) is sufficiently small.) Roughly speaking, the results are as follows: If \(\sigma<\lambda<\sigma_+\) and if \(a_+\) is sufficiently small, then the solution exists for all time. If \(\sigma<\lambda<\sigma_+\) and if \(a_+\) is sufficiently large, then the solution becomes infinite in finite time. For \(\lambda\geq\sigma_+\), the solution always becomes infinite in finite time. The key step is in analyzing the steady state problem. If \[ \Delta u +\lambda u+a(x)u^ p \quad\text{in } \Omega, \qquad u=0 \quad\text{on } \partial\Omega \] has a positive solution, then the solution of the original problem converges to the minimal solution of this one, and this minimal solution is the unique linearly stable steady-state solution. Otherwise, there is a metasolution of the problem \[ \Delta u +\lambda u- \max \{0,-a(x)\}u^p\quad \text{in } \Omega, \qquad u=0 \quad\text{on } \partial\Omega, \] that is, \(u\) satisfies the differential equation in some subdomain \(D\) of \(\Omega\) and vanishes on \(\partial D\cap \partial \Omega\) with \(u=\infty\) on \(\Omega\setminus D\). In this case, the solution of the parabolic problem converges to this metasolution in finite or infinite time.