L p -convergence rates to nonlinear diffusion waves for quasilinear equations with nonlinear damping

Decay of the 3D Quasilinear Hyperbolic Equations with Nonlinear Damping

“…The functions , 1 , and 2 present in (6) and (8) are material functions. The second law of thermodynamics imposes the restrictions that ( ) = 20 2 1 ( ) and 2 ( ) < 0, where the constant 20 comes from the Helmholtz free energy which has the form = 1 ( ) + (1/2) 20 P 2 .…”

“…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:…”

“…The authors in [6] removed restriction (11). Recently, the authors in [7] proved convergence rates for the half space problem.…”

“…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.…”

“…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. However, the antiderivative technique does not work in our high dimensional problem since the antiderivative technique is basically limited to onedimensional problem.…”