Maximal Time of Existence for $u_t+uu_x+u=0$

Question

Let $x\in \mathbb{R}$ and $t\geqslant0$. Consider the PDE $$u_t+uu_x+u=0 \qquad u(x,0)=h(x)\in C_0^\infty(\mathbb{R})$$ Show the maximal time of existence of the solution equals $\ln\Big(\min_{\sigma} \frac {h'(\sigma)}{1+h'(\sigma)}\Big)$

How would I go about showing for up to what time $t$ is the solution well defined? (I.e. the maximal time of existence).

I tried first to solve the above quasi-linear PDE using the method of characteristics, but was not able to solve it.

Any help would be much appreciated.

Answer 1

We follow the steps in this related post for $\alpha=1$. Thus, a shock wave issued from the initial position $x_0$ forms at the positive time $$ t = -\ln\left(1+\tfrac{1}{h'(x_0)}\right) = \ln\left(\tfrac{h'(x_0)}{1+h'(x_0)}\right) . $$ It remains to minimize the above expression w.r.t. $x_0$ to obtain the breaking time. Then, by using the fact that the logarithm is an increasing function, the desired result is obtained.