  We consider equations with the simplest hysteresis operator at the right-hand
side. Such equations describe the so-called processes "with memory" in which
various substances interact according to the hysteresis law.
  We restrict our consideration on the so-called "strong solutions" belonging
to the Sobolev class $W^{2,1}_q$ with sufficiently large $q$ and prove that in
fact $q=\infty$. In other words, we establish the optimal regularity of
solutions. Our arguments are based on quadratic growth estimates for solutions
near the free boundary.
