Stabilization of positive solutions for analytic gradient-like systems

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 inequality. Weighted 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.