I've been reading this pdf on Burger's equation: https://www.math.umd.edu/~mariakc/burgers.pdf and some other posts here as Breaking time of Burgers equation Consider $u_0\in C^1(\mathbb{R})$, strictly decreasing and with bounded derivative. All I find is that the breaking time is $$ T^*=\min_{x\in\mathbb R}\frac{-1}{u_0'(x)} $$ I understand the calculations: if we are given two different characteristics $X_x(t)=x+u_0(x)t, X_y(t)=y+u_0(y)t$, they intersect at $$t_{xy}=-\frac{x-y}{u_0(x)-u_0(y)}\overset{MVT}{=}\frac{-1}{u_0'(z_{xy})} $$ As I want the first instant ath which two characteristics intersect, I would take the infimum over $x,y\in\mathbb{R}$, which exists because the quotient is always $>0$. However, how do I know that it is in fact a minimum? Is it achieved? For now I only have $$ T^*=\inf_{x\in\mathbb R}\frac{-1}{u_0'(x)} $$
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