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

Authors : Mohammad SHAHROUZİ, Jorge FERREIRA

Pages : 208-216

Doi:10.33434/cams.941324

View : 15 | Download : 10

Publication Date : 2021-12-27

Article Type : Research Paper

Abstract :In this paper we consider $rinsert ignore into journalissuearticles values(x);-$Kirchhoff type equation with variable-exponent nonlinearity of the form $$ u_{tt}-\Delta u-\biginsert ignore into journalissuearticles values(a+b\int_{\Omega}\frac{1}{rinsert ignore into journalissuearticles values(x);}|\nabla u|^{rinsert ignore into journalissuearticles values(x);}dx\big);\Delta_{rinsert ignore into journalissuearticles values(x);}u+\beta u_{t}=|u|^{pinsert ignore into journalissuearticles values(x);-2}u, $$ associated with initial and Dirichlet boundary conditions. Under appropriate conditions on $rinsert ignore into journalissuearticles values(.);$ and $pinsert ignore into journalissuearticles values(.);$, 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