Universit´ e Fran¸cois Rabelais 2016 Laboratoire de Math´ ematiques et de Physique Th´ eorique Yves BELAUD (Universit´e de Tours, France) & Anh DAO NGUYEN (Ton Duc Thang University, Vietnam)

Extinction of solutions of semilinear high order parabolic equations with degenerate absorption potential In this paper, the authors investigate the extinction in a finite time for solutions of the nonlinear parabolic partial differential equation : (1)

ut + Lu + a(x) uq = 0

in Ω × (0, ∞),

for the Dirichlet Boundary Conditions in Ω a smooth bounded domain where 0 < q < 1 and L a divergent form operator of order 2m and a(x) ≥ 0 a.e.. Under some conditions on a(x), all the solutions of equation (1) vanish in a finite time. This work follows a long list of articles. Here, the main point is that they deal with high order operators.

References :

(1) Extinction of solutions of semilinear high order parabolic equations with degenerate absorption potential, Y. Belaud and A. Shishkov, J. Evol. Equ. 10, 857-882, (2010). (2) Long time vanishing property of solutions of some semilinear parabolic equations, Y. Belaud, B. Helffer & L. V´eron, Ann. I.H.P. Anal. Nonlin´eaire 18, 43-68 (2001). (3) Asymptotic behaviour of solutions of some nonlinear parabolic or elliptic equations, V.A. Kondratiev & L. V´eron, Asymptotic Analysis 14, 117-156 (1997). (4) Long-time extinction of solutions of some semilinear parabolic equations, Y. Belaud and A. Shishkov, J. Differential Equations 238, 64-86 (2007).

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