The general solution of the heat equation $\left\{\begin{array}{rcl} \partial_tu-\Delta u &=& 0\\ u(x,0)&=& f \end{array} \right.$ is given by $u(x,t)=\int\limits_{\mathbb R^n}\Phi(x-y,t)f(y)\mathrm dy$ with the fundamental solution $\Phi$ (wikipedia).
So why is the solution $u\in C^0([0,\infty)\times\mathbb R^n)\cap C^\infty((0,\infty)\times\mathbb R^n)$ bounded if f is bounded?