Proof that the solution of the heat equation is unique

[mathmari]

Hey!

I haven't really understood the following proof that the solution of the heat equation is unique. Could you explain it to me?

Heat equation with Dirichlet boundary conditions:

$$\left.\begin{matrix}u_t=u_{xx}, 0<x<L, t>0\\
u(0,t)=u(L,t)=0, t>0\\
u(x,0)=f(x)=0, 0<x<L\end{matrix}\right\}(1)$$
We want to show that the solution of this problem is unique:We suppose that the problem has two solutions, $u_1(x,t), u_2(x,t)$:

$$w(t)=\frac{1}{2}\int_0^L{|u(x,t)|^2}dx, t>0, (2)$$

$$u(x,t)=u_1(x,t)-u_2(x,t)$$
$$w(t)>0, (3)$$
$$w'(t)=\frac{1}{2} \int_0^L{(u_t u^*+u u^*_t)}dx$$
$$(1):w'(t)\frac{1}{2} \int_0^L{(u_{xx} u^*+u u^*_{xx})}dx$$
$$\int_0^L{u_{xx}u^*}dx=u_xu^*|_0^L-\int_0^L{u_xu^*_x}dx\overset{(1)}{=} - \int_0^L{|u_x|^2}dx$$
$$w'(t)=-\int_0^L{|u_x|^2}dx \leq 0, (4)$$
We know that $u_1(x,0)=u_2(x,0)=f(x), (5)$

So $u(x,0)=u_1(x,0)-u_2(x,0)=0$

$$w(t)=w(0)+\int_0^t{w'(s)}ds \leq 0, (6)$$
$$(3),(6) \Rightarrow w(t)=0, \forall t \geq 0$$
$$u_1=u_2$$First of all, why do we have to take at the beginning that $$w(t)=\frac{1}{2}\int_0^L{|u(x,t)|^2}dx, t>0, $$?? (Wondering)

 

[gtoguy97]

The proof is showing that the solution of the heat equation is unique by supposing that there are two solutions and then deriving a contradiction. The goal is to show that $u_1 = u_2$ which implies that the solution is unique. To start, we look at the equation $$w(t)=\frac{1}{2}\int_0^L{|u(x,t)|^2}dx, t>0$$ which is essentially the integral of the square of the absolute value of the solution $u(x,t)$. This is the quantity we want to examine in order to prove that $u_1=u_2$. We differentiate this equation with respect to $t$ to get $$w'(t)=\frac{1}{2} \int_0^L{(u_t u^*+u u^*_t)}dx$$ which gives us an expression for the rate of change of $w(t)$.We can then substitute the heat equation in for $u_t$ and $u^*_t$ and make use of the boundary conditions to simplify the expression. After some rearranging we obtain $$w'(t)=-\int_0^L{|u_x|^2}dx \leq 0$$ This tells us that the rate of change of $w(t)$ is negative or zero and therefore $w(t)$ decreases over time. We can then integrate this expression from $0$ to $t$ to get $$w(t)=w(0)+\int_0^t{w'(s)}ds \leq 0$$ where we have used the initial condition $u(x,0)=f(x)$ which implies that $w(0)=0$. Comparing this result with the original equation, we see that if $w(t)$ is always less than or equal to $0$, then it must be equal to $0$. This implies that $u_1=u_2$ since $u=u_1-u_2$. Therefore, the solution of the heat equation is unique.