TY  - JOUR
T1  - A Nonlinear $r(x)$-Kirchhoff Type Hyperbolic Equation: Stability Result and Blow up of Solutions with Positive Initial Energy
AU  - Shahrouzi, Mohammad
AU  - Ferreıra, Jorge
PY  - 2021
DA  - December
Y2  - 2021
DO  - 10.33434/cams.941324
JF  - Communications in Advanced Mathematical Sciences
PB  - Emrah Evren KARA
WT  - DergiPark
SN  - 2651-4001
SP  - 208
EP  - 216
VL  - 4
IS  - 4
LA  - en
AB  - 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.
KW  - Kirchhoff equation
KW  - stability result
KW  - variable exponents
KW  - blow-up
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UR  - https://doi.org/10.33434/cams.941324
L1  - https://dergipark.org.tr/en/download/article-file/1783056
ER  -