We consider the semilinear parabolic  equation $u_t= \Delta u+u^p$ on $\mathbb R^N$, where the power nonlinearity is subcritical. We first address the question of existence of entire solutions, that is, solutions defined for all $x\in\mathbb R^N$ and $t\in\mathbb R$. Our main result asserts that there are no positive radially symmetric bounded entire solutions. Then we consider radial solutions of the Cauchy problem. We show that if such a solution is global, that is, defined for all $t\ge 0$, then it necessarily converges to 0, as $t\to\infty$, uniformly with respect to $x\in\mathbb R^N$.