Long-time extinction of solutions of some semilinear parabolic equations

Yves Belaud (1) , Andrey Shishkov (2)

1 IDP - Institut Denis Poisson
2 IAMM - Institute of Applied Mathematics and Mechanics of NAS of Ukraine

We study the long-time behavior of solutions of semilinear parabolic equation of the following type ∂t u − ∆u + a0(x)u^q = 0 where a0(x) ≥ d0 exp(− [ω(|x|)]/|x|2 ), d0 > 0, 1 > q > 0, and ω is a positive continuousradial function. We give a Dini-like condition on the function ω by two different methods which implies that any solution of the above equation vanishes in a finite time. The first one is a variant of a local energy method and the second one is derived from semi-classical limits of some Schrödinger operators.

Mots clés: Nonlinear equation; Energy method; Vanishing solutions; Semi-classical analysis

Domaines: Equations aux dérivées partielles [math.AP]

Yves Belaud, Andrey Shishkov. Long-time extinction of solutions of some semilinear parabolic equations. Journal of Differential Equations, 2007, 238 (1), pp.64-86. ⟨10.1016/j.jde.2007.03.015⟩. ⟨hal-03922141⟩