A sufficient condition for global existence of solutions to one-dimensional hyperbolic free boundary problems (Q2706022)

A free boundary problem \(Tu_{xx}-\rho u_{tt}=0\) in \((0,+\infty)\times \{t>0\}\cap \{u>0\}\) and \((1/2)Tu_x^2-(1/2)\rho u_t^2=Q/(2T) \) on \((0,+\infty)\times \{t>0\}\cap \partial \{u>0\}\) with an initial condition \(u(x,0)=e(x)\), \(u_t(x,0)=g(x)\) in \([0,l_0)\) \((l_0>0)\) and \(u(x,0)=u_t(x,0)=0\) otherwise, and a boundary condition \(u(0,t)=f(t)\) (\(t>0\)) is investigated by a numerical method. Such a problem arises in some physical models describing a ``peeling-off'' phenomenon. The author makes use of the fixed domain method to resolve numerically the problem under consideration. The numerical computations suggest that the peeling speed, as a boundary condition, plays an important role in the existence of global solutions. Moreover, a sufficient condition for the existence of a global solution of the considered problem is proposed.