Asymptotic Behavior of Large Weak Entropy Solutions of the Damped p-system (Journal of Partial Differential Equations, 2020)
The asymptotic behavior of solutions of the damped p-system is known to be described by a nonlinear diffusion equation. The previous works on this topic concern either the case of small smooth data where estimates of high-order derivatives are available by energy methods or the case of special and small rough data. For large data, the existence of solutions is proved by using the method of compensated compactness. Thus the above mentioned energy estimates are not expected. However, the compensated compactness gives a very weak justification (in the mean in time) of the asymptotics. In the present paper we prove that the natural energy estimates, which does not involve derivatives, combined with this “convergence in the mean”, gives the strong convergence in L^p_{loc} space (p is finite) as expected.
Keywords: Asymptotic behavior; weak entropy solution; damped p-system