Vacuum energy decay from a q-bubble

We consider a finite-size spherical bubble with a nonequilibrium value of the $q$-field, where the bubble is immersed in an infinite vacuum with the constant equilibrium value $q_0$ for the $q$-field (this $q_0$ has already cancelled an initial cosmological constant). Numerical results are presented for the time evolution of such a $q$-bubble with gravity turned off and with gravity turned on. For small enough bubbles and a $q$-field energy scale sufficiently below the gravitational energy scale $E_{\text{Planck}}$, the vacuum energy of the $q$-bubble is found to disperse completely. For large enough bubbles and a finite value of $E_{\text{Planck}}$, the vacuum energy of the $q$-bubble disperses only partially and there occurs gravitational collapse near the bubble center.