Refined wave breaking for the one-dimensional nonlinear shallow water equations

This paper aims to give a refined wave breaking description of the Cauchy problem to the one-dimensional nonlinear shallow water equations providing a sharp estimate of the lifespan of the solutions depending on the amplitude and topography parameters, under a non-cavitation condition which excludes the scenario that the solutions have compact support. We construct smooth initial data with finite ${\dot{H}}^5$-norm such that the $L^{\mathrm{\infty }}$-norm of the spatial derivative of the solution blows up at one single point in finite time with a precise blowup profile.