Initial boundary value problem for the 3D quasilinear hyperbolic equations with nonlinear damping

ABSTRACT

We investigate global existence and asymptotic behavior of the 3D quasilinear hyperbolic equations with nonlinear damping on a bounded domain with slip boundary condition, which describes the propagation of heat waves for rigid solids at very low temperature, below about 20 K. The global existence and uniqueness of classical solutions are obtained when the initial data are near its equilibrium. Time asymptotically, the internal energy is conjectured to satisfy the porous medium equation and the heat flux obeys the classical Darcy’s-type law. Based on energy estimates, we show that the classical solution converges to steady state exponentially fast in time. Moreover, we also verify that the same is true for the corresponding initial boundary value problem of porous medium equation and thus justifies the validity of Darcy’s-type law in large time.

KEYWORDS:

• Darcy’s-type law
• quasilinear hyperbolic equations
• nonlinear damping
• classic solution

AMS SUBJECT CLASSIFICATIONS:

• 35B40
• 35L45
• 35L60
• 74D10

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was partially supported by the Hunan Provincial Natural Science Foundation of China [grant number 2017JJ2105]; the National Natural Science Foundation of China [grant number 11771150], [grant number 11571280], [grant number 11301172], [grant number 11226170]; Aid program for Science and Technology Innovative Research Team in Higher Educational Instituions of Hunan Province.