Formation of singularities for a family of 1D quasilinear wave equations

Abstract

We consider the blow-up of solutions to the following parameterized nonlinear wave equation: utt = c(u)uxx + λc(u)c(u)(ux) with the real parameter λ. In previous works, it was reported that there exist finite time blow-up solutions with λ = 1 and 2. However, the construction of a blow-up solution depends on the symmetric structure of the equation (e.g., the energy conservation law). In the present paper, we extend the blow-up result with λ = 1 to the case with λ ∈ (0, 1] by using a new L2/λ estimate. Moreover, some properties for the blow-up solution including the Hölder continuity are also discussed.