On stabilization of the solutions of parabolic equations with small parameter (original) (raw)

Stabilization of solutions of certain parabolic equations and systems

Mathematical Notes of the Academy of Sciences of the USSR, 1968

This paper concerns the investigation of the stabilization of solutions of the Cauchy problem for a system of equations of the form ~u/at = 5u + Fl(u , v); 0v/St = ~v + F2~ , v). It is proved that under certain assumptions the behavior of solutions as t -* | is determined by mutual arrangement of the set of initial conditions { (u, v): u = fl(x), v = f2(x), x~Rn} and the trajectories of the system of ordinary differential equations du/dt = Ft(u , v), dv/dt = F 2 (u, v). The question of stabilization of the solutions of a single quasilinear parabolic equation is also considered.

Stability results for a class of non-linear parabolic equations

Ann Mat Pur Appl, 1977

We study the asymptotic behaviour o] the solutions o/ the equation ut = Au + + ~u-lu]~u. Denoting by ~o the principal eigenvalue o] the second.order di]ferential operator A, we shall prove that if ). ~ )~ the only equilibrium solution, namely zero, is asymptotically stable, whereas, i] 2 > 20, the nontrivial equilibrium solutions without internal zeros are asymptotically stable. Attraetivity and stability are proved both in the L~.norm and in the H~-norm.

The Stability of the Solutions for a Quasilinear Degenerate Parabolic Equation

arXiv (Cornell University), 2019

The equation arising from Prandtl boundary layer theory ∂u ∂t − ∂ ∂x i a(u, x, t) ∂u ∂x i − f i (x)D i u + c(x, t)u = g(x, t) is considered. The existence of the entropy solution can be proved by BV estimate method. The interesting problem is that, since a(•, x, t) may be degenerate on the boundary, the usual boundary value condition may be overdetermined. Accordingly, only dependent on a partial boundary value condition, the stability of solutions can be expected. This expectation is turned to reality by Kruzkov's bi-variables method, a reasonable partial boundary value condition matching up with the equation is found first time. Moreover, if a xi (•, x, t) | x∈∂Ω = a(•, x, t) | x∈∂Ω = 0 and f i (x) | x∈∂Ω = 0, the stability can be proved even without any boundary value condition.

Large time behaviour of solutions to a class of non-autonomous, degenerate parabolic equations

Mathematische Annalen, 2010

Large time behaviour of solutions to a class of non-autonomous, degenerate parabolic equations F. Ragnedda · S. Vernier Piro · V. Vespri Abstract We consider a class of non-autonomous, degenerate parabolic equations and we study the asymptotic behaviour of the solutions. Even if the equation depends explicitly upon the time, we prove that several asymptotic properties, valid for the autonomous case, are preserved in this more general situation. To our knowledge, it is the first time that the asymptotic behaviour of solutions to non-autonomous equations is studied.