Let $u=u(x,t)$ be the solution of the cauchy problem $\frac{\partial u}{\partial t}+\big({\frac{\partial u}{\partial x}}\big)^2 = 1$,
where $x \in \mathbb{R}$ , $t\gt0$ and $u(x,0)= -x^2$.
A. $u(x,t)$ exists for all $x \in \mathbb{R}$ and $t>0$.
B. ${|u(x,t)|\to \infty}$ as ${t \to t_0}$ for some $t_0>0$ and $x\neq0$.
C. $u(x,t)\leq0$ for all $x \in \mathbb{R}$ and for all $t\lt1/4$.
D. $u(x,t)\gt0$ for some $x \in \mathbb{R}$ and $0\lt t \lt\ 1/4$.
My approach:
The above pde is non-linear in $p$ and $q$, where $p=\frac{\partial u}{\partial x}$ and $q=\frac{\partial u}{\partial t}$. So, the above pde is $p^2+q=1$.
Applying charpit's method, the solution is given by $u(x,t) = ax +(1-a^2)t +b$. After applying the given intial condition the solution reduces to $u(x,t) = (1-a^2)t - x^2$. Now my first question is how to get the value of $a$ and then how to verify the given four options.