Asymptotics of Solutions of a Perfect Fluid Coupled with a Cosmological Constant in Four-Dimensional Spacetime with Toroidal Symmetry

Asymptotics of solutions of a perfect fluid when coupled with a cosmological constant in four-dimensional spacetime with toroidal symmetry are studied. In particular, it is found that the problem of self-similar solutions of the first kind for a fluid with the equation of state, $p=k\rho$, can be reduced to solving a master equation of the form, $$2F(q,k)\frac{q^{\mathrm{\prime }\mathrm{\prime }}(\xi )}{q^{\mathrm{\prime }}(\xi )}-G(q,k)q^{\mathrm{\prime }}(\xi )=\frac{4}{\xi }\mathrm{.}$$ For $k=0$ and $k=-1\mathrm{/}3$ the general solutions are obtained and their main local and global properties are studied in detail.