Asymptotic behavior of solutions to quasilinear hyperbolic equations with nonlinear damping

Decay of the 3D Quasilinear Hyperbolic Equations with Nonlinear Damping

“…More precisely, we prove global existence and uniqueness of strong solutions when the initial data is near its equilibrium in the sense of 3 -norm. Moreover, if, additionally, -norm (1 ≤ < 6/5) of the initial perturbation is finite, we also show the optimal -2 decay rates for the solutions without the additional technical assumptions for the nonlinear damping (V) given by Li and Saxton in [5].…”

“…Based on the system's special dissipation structure and delicate analysis and interpolation technique on the nonlinear terms, the desired energy estimates can be achieved. Compared to the one-dimensional results in [5][6][7], the approach is new and quite different here. In [5][6][7], the antiderivative technique plays an essential role in proving their main results.…”

“…the existence and asymptotic behavior of smooth solutions for the Cauchy problem and initial boundary value problem have been considered by [5,6] and [7], respectively. In [5], they obtained the convergence rates of the smooth solutions for the Cauchy problem under the following technical assumptions for the nonlinear damping:…”

“…In [5], they obtained the convergence rates of the smooth solutions for the Cauchy problem under the following technical assumptions for the nonlinear damping:…”

“…Equation (7), related to (4) via (5), is however linear and does not fully describe the properties of heat propagation in solids (cf. [1][2][3][4] and references therein).…”