Past exam of the algebra, analysis and differential equations 1 course of the University of Oxford / 2004 / Differential equations / 9

This is the heat equation, with $y$ as time. The stated zero solution is unique. Define $E(y)=\frac{1}{2}{\int }_0^{1}u(x,y{)}^{2}dx$. For $0<y<1$, integration by parts and $u_y=u_{xx}$ giveThe 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 $u,u_x,u_y$.

This is Laplace's equation. Zero is not unique: with $k=\pi /2$,is nonzero and harmonic. It vanishes on $y=0$ and $x=0$, while $u_x(1,y)=k\cos (k)\sinh (ky)=0$.

A sufficient additional condition is $u(x,1)=0$ on the top edge. Green's first identity then givesOn the bottom, left and top edges $u=0$; on the right edge the normal derivative is zero. Thus the gradient vanishes and $u$ is constant; the bottom value makes that constant zero. For $C^2(D)\cap C^1(\overline{D})$, apply the identity on inset rectangles and pass to the boundary using continuity of the function and its gradient.

This is the wave equation, again taking $y$ as time. Zero is not unique: for $k=\pi /2$,satisfies $u_{xx}=u_{yy}$ and all the stated boundary conditions. The initial displacement is zero but the initial velocity is not prescribed.

A sufficient extra condition is $u_y(x,0)=0$. For the resulting homogeneous data use wave energyThe equation and integration by parts give $E^{\prime }(y)=[u_x{u}_y{]}_0^1=0$. At $x=1$ the factor $u_x$ vanishes, and at $x=0$ the factor $u_y$ vanishes because $u(0,y)=0$. The initial displacement implies $u_x(x,0)=0$, so the extra initial velocity condition gives $E(0)=0$. Consequently $u_x=u_y=0$ and $u=0$. With the stated boundary regularity, integrate the local energy conservation law over inset rectangles and let their edges approach the boundary; the energy and flux terms converge by continuity of the first derivatives.