The blow-up curve of solutions to one dimensional nonlinear wave equations with the Dirichlet boundary conditions (Q2300969)

It is known that, under rather general conditions, solutions of equations of the form \(u_{tt}-u_{xx}=f(u)\) on the line may breakdown \(+\infty\) along points of a blow-up set. The relation between the regularity properties of the solution and that of the ``singularity data'' that characterize the blow-up behavior is not well-understood. For more general equations of the form \(u_{tt}-u_{xx}=f(u,u_x,u_t)\), singularities that ``do not propagate'' may occur (see [\textit{S. Kichenassamy}, Fuchsian reduction. Applications to geometry, cosmology and mathematical physics. Basel: Birkhäuser (2007; Zbl 1169.35002)], Rem.~10.37). This paper addresses an equation of the latter form, namely \(u_{tt}-u_{xx}=2^p|u_t|^{p-1}u_t\) on the half-line \((x>0)\), with boundary condition \(u_t(0,t)=0\), and \(p>5\), assuming technical conditions [(A1)--(A7), p.~342--343] on the Cauchy data, involving up to three derivatives. It shows that solutions with such data exhibit gradient blow-up for all positive \(x\) near 0, and that the equation of the blow-up curve (that is, the first singularity datum) fails to be spacelike up to the boundary. The solution may be extended to the whole line by reflection. The authors' main point is that such results may be obtained for equations that lack the variational structure that plays a central role in some earlier work.