Exact bounds for derivatives of classical solution of Cauchy's problem for a quasilinear equation (Q581750)

P. Lax proved that the Cauchy problem \[ (1)\quad \partial u(x,t)/\partial t+a(u)\partial u(x,t)/\partial x=0,\quad x\in R^ 1,\quad t\geq 0;\quad (2)\quad u(x,0)=u_ 0(x) \] has no classical solution for large t [see \textit{P. D. Lax}, Am. Math. Monthly 79, 227-241 (1972; Zbl 0228.35019)]. On the other hand, for small \(t>0\) classical solution always exists. In this paper the author shows that u(x,t) is implicitly determined in the form \(u(x,t)=u_ 0(x-ta(u))\) with exact bounds for the derivatives \(\partial u(x,t)/\partial x\), \(\partial u(x,t)/\partial t\) in \(t\in [0,T]\) for some T. Moreover in the case \(T=\infty,\partial u(x,t)/\partial x\) uniformly converges to zero as \(t\to \infty\), while in the case \(T<\infty\) the derivative converges to \(\infty\) as \(t\to T-0\) at some point x, i.e. there is no classical solution for \(t>T\).