A Nonlinear $r(x)$-Kirchhoff Type Hyperbolic Equation: Stability Result and Blow up of Solutions with Positive Initial Energy

Year 2021, Volume: 4 Issue: 4, 208 - 216, 27.12.2021

Mohammad Shahrouzi , Jorge Ferreıra

https://doi.org/10.33434/cams.941324

Cited By: 1

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

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