Past exam of the algebra, analysis and differential equations II course of the University of Oxford / 2012 / 9 / a / Solution

For a quasilinear partial differential equation, the characteristic equations for a quasilinear equation are the ordinary differential equationsA solution surface is locally a differentiable graph $z=u(x,y)$ whose tangent plane contains the vector $(P,Q,R)$ at every point. Its normal is $(-u_x,-u_y,1)$, so the tangency condition is precisely $P{u}_x+Q{u}_y=R$. Thus every characteristic lying on that graph follows the vector field without leaving the graph.

Parameterize the given initial curve by $\gamma (t)=(x_0(t),y_0(t),z_0(t))$. For each $t$, solve the characteristic equations for a quasilinear equation with initial value $\gamma (t)$ at $s=0$. The resulting map $(s,t)\mapsto (x(s,t),y(s,t),z(s,t))$ sweeps out a candidate solution surface. If the coefficients and initial data are continuously differentiable, the Picard-Lindelöf theorem supplies this local family. To make it a graph we also need the projection $(s,t)\mapsto (x,y)$ to be locally invertible. At $s=0$ its Jacobian determinant isThe Noncharacteristic Cauchy data condition is $J(0,t)\ne 0$. The inverse function theorem then gives $s=s(x,y)$ and $t=t(x,y)$, and hence $u(x,y)=z(s(x,y),t(x,y))$. Its derivatives obey the PDE by the chain rule. The same characteristic initial-value uniqueness gives local uniqueness of this solution wherever the projection remains invertible and the relevant characteristic meets the prescribed data.