Critical exponent for global solutions in a fourth‐order hyperbolic equation of p‐Kirchhoff type

Abstract

We study a fourth‐order hyperbolic equation involving Kirchhoff type ‐Laplacian and superlinear source, subject to zero Navier boundary condition, where is an open bounded domain in with ; denotes the maximal existence time; and and are constants. For , using auxiliary function method and Sobolev inequality, we prove that there are only global solutions. For , we obtain the optimal classification of initial energy and Nehari energy, which guarantees the existence of blow‐up solutions and global solutions. In the critical case , we find out that the coefficients of the Kirchhoff term and the superlinear source play important role in separating out the property of weak solutions.