@article{article_941324, title={A Nonlinear $r(x)$-Kirchhoff Type Hyperbolic Equation: Stability Result and Blow up of Solutions with Positive Initial Energy}, journal={Communications in Advanced Mathematical Sciences}, volume={4}, pages={208–216}, year={2021}, DOI={10.33434/cams.941324}, author={Shahrouzi, Mohammad and Ferreıra, Jorge}, keywords={Kirchhoff equation, stability result, variable exponents, blow-up}, abstract={<div style="text-align:justify;">In this paper we consider $r(x)-$Kirchhoff type equation with variable-exponent nonlinearity of the form $$ u_{tt}-\Delta u-\big(a+b\int_{\Omega}\frac{1}{r(x)}|\nabla u|^{r(x)}dx\big)\Delta_{r(x)}u+\beta u_{t}=|u|^{p(x)-2}u, $$ associated with initial and Dirichlet boundary conditions. Under appropriate conditions on $r(.)$ and $p(.)$, stability result along the solution energy is proved. It is also shown that regarding arbitrary positive initial energy and suitable range of variable exponents, solutions blow-up in a finite time. </div>}, number={4}, publisher={Emrah Evren KARA}