Partial derivatives of a strong solution are also solutions?

[cjc]

Homework Statement

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.

Homework Equations

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

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! =]

 

[mighty2000]

Thank you for your post. To begin, we can rewrite the heat equation as follows:

u_{t}=\alpha^{2}u_{xx} = \frac{\partial u}{\partial t}=\alpha^{2}\frac{\partial^{2}u}{\partial x^{2}}

Now, using the theorem you mentioned, we can write:

\frac{\partial u}{\partial t} = \int_{\mathbb{R}} \alpha^{2}\frac{\partial^{2}u}{\partial x^{2}} dr

Since u is a strong solution to the heat equation, we know that the integral defining u is uniformly convergent. This also means that the integral defining \frac{\partial u}{\partial t} is uniformly convergent. Therefore, we can apply the theorem and conclude that \frac{\partial u}{\partial t} is differentiable and \frac{\partial^{2}u}{\partial x^{2}} is its derivative.

Similarly, we can write:

\frac{\partial^{2}u}{\partial x^{2}} = \int_{\mathbb{R}} \frac{\partial}{\partial x}(\alpha^{2}\frac{\partial u}{\partial x}) dr

Again, since u is a strong solution, the integral defining \frac{\partial^{2}u}{\partial x^{2}} is uniformly convergent. Therefore, we can apply the theorem and conclude that \frac{\partial^{2}u}{\partial x^{2}} is differentiable and \frac{\partial}{\partial x}(\alpha^{2}\frac{\partial u}{\partial x}) is its derivative, which is equal to \alpha^{2}u_{xx}.

Therefore, we have shown that both u_{t} and u_{x} are solutions to the heat equation. I hope this helps! If you have any further questions, please don't hesitate to ask.