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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?

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up vote 3 down vote accepted

The fundamental solution is positive, integrable on $\mathbb{R}^n$ and $\int_{\mathbb{R}^n}\Phi(x,t)\,dx=1$ for all $t>0$. Then $$ |u(x,t)|\le\int_{\mathbb{R}^n}\Phi(x-y,t)|f(y)|\,dy\le\sup_{y\in\mathbb{R}^n}|f(y)|\int_{\mathbb{R}^n}\Phi(x-y,t)\,dy=\sup_{y\in\mathbb{R}^n}|f(y)|<\infty. $$

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