On Multidimensional Axisymmetric Oscillations of a Collisional Cold Plasma

We study the influence of the friction term on the radially symmetricsolutions of the repulsive Euler-Poisson equations with a non-zero background,corresponding to cold plasma oscillations in many spatial dimensions. It isshown that for any arbitrarily small non-negative constant frictioncoefficient, there exists a neighborhood of the zero equilibrium in the $C^1$norm such that the solution of the Cauchy problem with initial data belongingto this neighborhood remains globally smooth in time. Moreover, this solutionstabilizes to zero as $t\to\infty$. This result contrasts with the situation ofzero friction, where any small deviation from the zero equilibrium generallyleads to a blow-up. Our method allows us to estimate the lifetime of smoothsolutions. Further, we prove that for any initial data, one can find suchcoefficient of friction that the respective solution to the Cauchy problemkeeps smoothness for all $t>0$ and stabilizes to zero. We also present theresults of numerical experiments for physically reasonable situations, whichallows us to estimate the value of the friction coefficient, which makes itpossible to suppress the formation of singularities of solutions.