Geodesic
Asymptotic stability of solutions to quasilinear damped wave equations with variable sources
Izvestiya. Mathematics , Tome 88 (2024) no. 4, pp. 794-814

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In this paper, we consider the following quasilinear damped hyperbolic equation involving variable exponents: $$ u_{tt}-\operatorname{div}\bigl( |\nabla u|^{r(x)-2}\nabla u\bigr)+|u_t|^{m(x)-2} u_t-\Delta u_t=|u|^{q(x)-2}u, $$ with homogenous Dirichlet initial boundary value condition. An energy estimate and Komornik's inequality are used to prove uniform estimate of decay rates of the solution. We also show that $u(x, t)=0$ is asymptotic stable in terms of natural energy associated with the solution of the above equation. As we know, such results are seldom seen for the variable exponent case. At last, we give some numerical examples to illustrate our results.
Keywords: Komornik inequality, $r(x)$-Laplacian operator, damped quasilinear, variable exponent. 
@article{IM2_2024_88_4_a6,
     author = {Xiaoxin Yang and Xiulan Wu and Jibao Zhuang},
     title = {Asymptotic stability of solutions to quasilinear damped wave equations with variable sources},
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     pages = {794--814},
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     year = {2024},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2024_88_4_a6/}
}
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Xiaoxin Yang; Xiulan Wu; Jibao Zhuang. Asymptotic stability of solutions to quasilinear damped wave equations with variable sources. Izvestiya. Mathematics , Tome 88 (2024) no. 4, pp. 794-814. http://geodesic.mathdoc.fr/item/IM2_2024_88_4_a6/