Bounding a Subsolution of the Heat Equation

Question

As the title suggests, I'd like to bound a subsolution of the heat equation. I have \begin{align*} u_t - \Delta u &\le 0 \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text{ in } \mathbb R^n \times (0,\infty) \\ u(x,0) &= g(x) \,\,\,\,\,\,\,\,\, \text{in } \mathbb R^n\end{align*} and I would like to claim that $$u(x,t) \le \int_{\mathbb R^n} \Phi(x-y,t) g(y) \,dy.$$ Is that claim true? It seems intuitive that it should be true, but the reasoning as to why is eluding me. Any help would be appreciated.