= Solution
This is the \b[<heat equation>], with $y$ as time. The stated zero solution is \b[unique]. Define $E(y)=\frac12\int_0^1u(x,y)^2\,dx$. For $0<y<1$, <integration by parts> and $u_y=u_{xx}$ give
$$
E'(y)=\int_0^1u\,u_{xx}\,dx=[uu_x]_0^1-\int_0^1u_x^2\,dx=-\int_0^1u_x^2\,dx\leq0.
$$
The boundary term is zero because $u(0,y)=0$ and $u_x(1,y)=0$. Also $E(0)=0$. Nonnegativity forces $E(y)=0$, and <continuity> gives $u=0$ throughout the square. The unknown final-time boundary values are determined by evolution and need not be prescribed. Under the stated regularity the integrations can first be made on an interior interval and then passed to the endpoints using <continuity> of $u,u_x,u_y$.
Back to article page
