Blow up solution of inverse problem for nonlinear hyperbolic equation with variable-exponents

Abstract

In this work, we study the following inverse problem with variable exponents:
utt − div(|∇u|^(r(.)−2)∇u) + a|ut|^(m(.)−2)ut − b|u|^(p(.)−2)u = f (t)w(x),
where a, b > 0 are constants and variable exponents p(.), r(.) and m(.) are given functions. We analyzed the finite-time blow-up of the solution using the alternative method proposed by Georgiev and Todorova with negative initial energy.