The long-time dynamical properties of an arbitrary positive solution $u(t)$, $t\ge 0$,
to autonomous gradient-like systems are investigated.
These evolutionary systems are generated by semilinear parabolic Dirichlet problems where coefficients and
nonlinearities are allowed to be unbounded near the
boundary $\partial \Omega$ of the underlying
bounded domain $\Omega\subset \mathbb R^N$.
Analyticity of the potential is used to show that every
positive solution of the system asymptotically approaches a (single) steady-state solution.
A key tool in the proof is a Lojasiewicz-Simon-type
Lebesgue and Sobolev spaces are employed. Important applications include the nonlinear
heat and porous medium equations that contain nonlinearities which are not
necessarily analytic on the boundary of the domain.