- Communications in Advanced Mathematical Sciences
- Vol: 4 Issue: 4
- A Nonlinear $r(x)$-Kirchhoff Type Hyperbolic Equation: Stability Result and Blow up of Solutions wit...
A Nonlinear $r(x)$-Kirchhoff Type Hyperbolic Equation: Stability Result and Blow up of Solutions with Positive Initial Energy
Authors : Mohammad Shahrouzi, Jorge Ferreira
Pages : 208-216
Doi:10.33434/cams.941324
View : 10 | Download : 6
Publication Date : 2021-12-27
Article Type : Research
Abstract :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.Keywords : Kirchhoff equation, stability result, variable exponents, blow-up