Behavior of solutions to nonlinear parabolic equations at large time
Initial-boundary value problems for systems of quasilinear parabolic equations are considered. The couple of the equations is assumed to be cooperative, so that the resulting semiflow falls in the class of strongly monotone dynamical systems. The latter means that any two pointwise and componentwise ordered initial conditions give solutions that are ordered at any time instant. The author presents several theorems on existence of stable and unstable equilibria, domains of attraction, existence of monotone heteroclinic solutions, eventual monotonicity and stabilization of solutions. The results presented overlap several theorems in the existing literature. Some of them are quoted in the paper under review, for others see the monographs [\textit{P. Hess}, Periodic-parabolic boundary value problems and positivity, Pitman Res. Notes 247, Longman, Harlow (1991; Zbl 0731.35050)] and [\textit{H. L. Smith}, Monotone dynamical systems, American Mathematical Society, Providence, RI (1995; Zbl 0821.34003)].
scientific article; zbMATH DE number 923517
scientific article