$L^1$-Convergence to Generalized Barenblatt Solution for Compressible Euler Equations with Time-Dependent Damping

The large time behavior of entropy solution to the compressible Euler equations for polytropic gas (the pressure $p(\rho)=\kappa\rho^{\gamma}, \gamma>1$) with dependent damping like $-\frac{1}{(1+t)^\lambda}\rho u$ ($0<\lambda<1$) is investigated. By introducing an elaborate iterative method and using intensive analysis, it proved that $L^\infty$ finite initial mass converges strongly in natural $L^1$ topology a fundamental porous media equation (PME) time-dependent diffusion, called by generalized Barenblatt solution. It interesting decay rate getting faster as $\lambda$ increases $(0, \frac{\gamma}{\gamma+2}]$, while slower $[ \frac{\gamma}{\gamma+2}, 1)$.