# Partial derivatives of a strong solution are also solutions?

1. Feb 23, 2013

### cjc

1. The problem statement, all variables and given/known data

For the heat equation $u_{t}=\alpha^{2}u_{xx}$ for $x\in\mathbb{R}$ and $t>0$, show that if $u(x,t)$ is a strong solution to the heat equation, then $u_{t}$ and $u_{x}$ are also solutions.

2. Relevant equations

$u_{t}=\alpha^{2}u_{xx}$

3. The attempt at a solution

I've considered the following theorem which shows that the integral defining $u$ and the integrals of partial derivatives are all uniformly convergent. It goes as follows:

Let $g(r,s)$ be defined and continuous on $\mathbb{R}\times I$, where $I$ is an interval, and suppose $\partial g/ \partial s$ exists and is continuous. If the improper integral for $G(s)=\int_{\mathbb{R}} g(r,s) dr$ and the improper integral $\int_{\mathbb{R}}\frac{\partial g}{\partial s}(r,s)dr$ are uniformly convergent on $I$, then $G$ is differentiable and $G'(s)=\int_{\mathbb{R}}\frac{\partial g}{\partial s}(r,s)dr$.

My question would be how do I begin to apply this to the given problem?

Thank you so much! =]