Stabilization of positive solutions for analytic gradient-like systems
Peter Takáč · Discrete and Continuous Dynamical Systems · 2000
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 andnonlinearities are allowed to be unbounded near the boundary $\partial \Omega$ of the underlyingbounded domain $\Omega\subset \mathbb R^N$. Analyticity of the potential is used to show that everypositive solution of the system asymptotically approaches a (single) steady-state solution. A key tool in the proof is a Lojasiewicz-Simon-type inequality. WeightedLebesgue and Sobolev spaces are employed. Important applications include the nonlinear heat and porous medium equations that contain nonlinearities which are notnecessarily analytic on the boundary of the domain.