A Logarithmic Decay of the Energy for the Hyperbolic Equation with Supercritical Damping (Journal of Partial Differential Equations, 2024)
We are concerned with the following quasilinear wave equation involving variable sources and supercritical damping: $$u_{tt}-{\rm div}(| abla u|^{p(x)-2} abla u)+|u_t|^{m-2}u_t=|u|^{q(x)-2}u.$$ Generally speaking, when one tries to use the classical multiplier method to analyze the asymptotic behavior of solutions, an inevitable step is to deal with the integral $\int_Ω |u_t|^{m−2}u_tudx.$ A usual technique is to apply Young’s inequality and Sobolev embedding inequality to use the energy function and its derivative to control this integral for the subcritical or critical damping. However, for the supercritical case, the failure of the Sobolev embedding inequality makes the classical method be impossible. To do this, our strategy is to prove the rate of the integral $\int_Ω |u|^ mdx$ grows polynomially as a positive power of time variable $t$ and apply the modified multiplier method to obtain the energy functional decays logarithmically. These results improve and extend our previous work [12]. Finally, some numerical examples are also given to authenticate our results.