Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

share|improve this question

1 Answer 1

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 $$ |u(x,t)|\le\int_{\mathbb{R}^n}\Phi(x-y,t)|f(y)|\,dy\le\sup_{y\in\mathbb{R}^n}|f(y)|\int_{\mathbb{R}^n}\Phi(x-y,t)\,dy=\sup_{y\in\mathbb{R}^n}|f(y)|<\infty. $$

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.