 Solution

The method of characteristics usesFor a graph parametrized by $(x,y,u(x,y))$, a normal vector is $N=(-u_x,-u_y,1)$. The PDE says $N\cdot (a,b,c)=0$, so the characteristic vector is tangent to the solution surface. Flowing a suitable initial curve along these directions constructs a surface; where its projection to the $(x,y)$ plane has nonzero Jacobian determinant, the inverse function theorem makes it a graph. Local uniqueness of this general flow needs the usual regularity assumptions beyond mere continuity. The particular characteristic system below is smooth and can be solved explicitly.

Label the initial points by $\eta \ge 0$. Since $dx/ds=1$, take $x=s$. The other two equations integrate toEliminating $\eta =y-xu+x^3/3$ gives the implicit solutionFor each fixed $|x|<1$, the derivative of the displayed $y$ with respect to $\eta$ is $1+x\cos \eta \ge 1-|x|>0$. It increases from its value at $\eta =0$ to infinity, so it is a bijection from $[0,\infty )$ onto $[y_L(x),\infty )$, whereThus there is exactly one characteristic label, and hence one solution value, throughout the requested region, including the initial-label boundary as a trace.

The characteristic fold for sinusoidal data in a forced Burgers equation is seen by differentiating with respect to $y$:For a fixed integer $k\ge 0$, let $y_{\ast }=(2k+1)\pi +7/6$. As $x\to 1^{-}$ at this fixed $y_{\ast }$, the label tends to $(2k+1)\pi$, giving $u_y\to -\infty$. To justify the label limit, the limiting map at $x=1$ is strictly increasing despite isolated zero derivatives, so its inverse has a unique continuous limit. Similarly, at any fixed $y_{\ast }=2k\pi -7/6$ with $k\ge 1$, the limit as $x\to -1^{+}$ has $\eta \to 2k\pi$ and $u_y\to +\infty$. These fixed values lie within the region near the respective boundary, proving the requested gradient blow-up.