I found the PDE solution:
$$u_t+u_x=u^2$$ $$u(0,x)=\cos(x)$$
The solution is:
$$u(t,x)=\frac{\cos(x-t)}{1-t\cos(x-t)}$$
Now how do you find the lowest $ t $ to which the solution "explodes" and which point it "explodes". I tried to parse when $t\cos(x-t)$ tends $1$, but I could not find.
Hint: Write $x=r+t$ and put $f(r,t) = \tfrac{\cos r}{1-t\cos r}$. Then $f$ "explodes" when $1-t\cos r = 0$, i.e., when $\cos r = \tfrac1t$. You can write $u$ in terms of $f$.