Lifespan of solutions to a hyperbolic type Kirchhoff equation with arbitrarily high initial energy

Abstract

In this paper, an initial boundary value problem for a hyperbolic type Kirchhoff equation with a strong dissipation and a general nonlinearity is considered. First, local existence and uniqueness of weak solutions are obtained with the help of Banach fixed point theorem. Then, by constructing an auxiliary functional and adopting the concavity argument, we give a new finite time blow-up criterion for this problem, which in particular implies that the problem admits blow-up solutions with arbitrarily high initial energy. Meanwhile, a bound for the blow-up time is derived from above. Further, we obtain a lower bound for the blow-up time when blow-up occurs. From methods to results, we partially extend the ones obtained in earlier literature.