Asymptotic behavior of solutions to the Euler-Korteweg equations with time-dependent damping

Abstract

This paper is concerned with the asymptotic behavior of solutions to the Cauchy problem for the one-dimensional Euler-Korteweg equations with time-dependent damping effect $ -\frac{\mu }{(1+t)^{\lambda}}u $ for $ 0<\lambda<1 $, $ \mu>0 $. For the case that $ v_+\neq v_- $, by means of the time-weighted energy method, we prove that the solutions to the Cauchy problem exist uniquely and globally, and tend to the nonlinear diffusion waves time-asymptotically when the initial perturbations around the diffusion waves are small enough, and we also obtain the algebraic time-convergence-rates for the solutions toward the nonlinear diffusion waves. Moreover, for the case that $ v_+ = v_- $, we show that the solutions of the Cauchy problem of this model converge to the constant states in time algebraically provided the initial data are close to the constant states. This study generalizes the results in [H.-B. Cui, J. Du, J. Evol. Equ., 18(2018), 29-47] which considered the Euler-Korteweg equations with constant coefficient damping.