  We show that the multi-dimensional compressible Euler system for isothermal
flow of an ideal, polytropic gas admits global-in-time, radially symmetric
solutions with unbounded amplitudes due to wave focusing. The examples are
similarity solutions and involve a converging wave focusing at the origin. At
time of collapse, the density, but not the velocity, becomes unbounded,
resulting in an expanding shock wave. The solutions are constructed as
functions of radial distance to the origin $r$ and time $t$. We verify that
they provide genuine, weak solutions to the original, multi-d, isothermal Euler
system.
  While motivated by the well-known Guderley solutions to the full Euler system
for an ideal gas, the solutions we consider are of a different type. In
Guderley solutions an incoming shock propagates toward the origin by
penetrating a stationary and "cold" gas at zero pressure (there is no counter
pressure due to vanishing temperature near the origin), accompanied by blowup
of velocity and pressure, but not of density, at collapse. It is currently not
known whether the full system admits unbounded solutions in the absence of
zero-pressure regions. The present work shows that the simplified isothermal
model does admit such behavior.
