  We study the character of dependence on the data and the nonlinear structure
of the equation for the solutions of the homogeneous Dirichlet problem for the
evolution $p(x,t)$-Laplacian with the nonlinear source \[
u_t-\Delta_{p(x,t)}u=f(x,t,u),\quad (x,t)\in Q=\Omega\times (0,T), \] where
$\Omega$ is a bounded domain in $\mathbb{R}^n$, $n\geq 2$, and $p(x,t)$ is a
given function $p(\cdot):Q\mapsto (\frac{2n}{n+2},p^+]$, $p^+<\infty$. It is
shown that the solution is stable with respect to perturbations of the variable
exponent $p(x,t)$, the nonlinear source term $f(x,t,u)$, and the initial data.
We obtain quantitative estimates on the norm of the difference between two
solutions in a variable Sobolev space through the norms of perturbations of the
nonlinearity exponent and the data $u(x,0)$, $f$. Estimates on the rate of
convergence of a sequence of solutions to the solution of the limit problem are
derived.
