ABSTRACT The main goal of this paper is to explore why observations of many astrospheres (or circumstellar bubbles) show quite stable and smooth structures of astropauses – the tangential discontinuities separating the stellar and interstellar winds – while both theory and numerical simulations suggest that tangential discontinuities are unstable due to well known Kelvin–Helmholtz (K-H) instability. It was recognized before that magnetic fields may stabilize the astropauses. In this paper, we explore another mechanism to reduce the K-H instability of the astropauses. This mechanism is a periodic change of the stellar wind dynamic pressure. Fluctuations of the stellar wind parameters are quite expected. For example, the Sun has an 11-yr cycle of global activity although there are also shorter periods of the solar wind fluctuations. We performed the parametric numerical study and demonstrate that the development of the K-H instability depends on the dimensionless parameter χ, which is the ratio of the stellar wind terminal speed and interstellar flow speed. The larger the parameter χ, the larger the fluctuations caused by the K-H instability. It has been shown that the K-H instability is convective which agrees with the previous linear analysis. The stabilization of the astropause by the periodic fluctuations in the stellar wind lead is demonstrated. It is shown that for the solar wind the most effective stabilization occurs when the period of stellar parameter change is about 1–4 yr. For the 11-yr solar cycle, the stabilization effect is weaker.