´ Franc Universite ¸ ois Rabelais Tours / University of Sciences of Ho Chi Minh City M2 report in mathematics 2014/2015
Free boundary in some reaction-diffusion equation Yves BELAUD,
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Summary. Let Ω be a bounded domain in RN with a smooth boundary and b(x) a bounded and measurable nonnegative function. If 0 < q < 1, Belaud, Helffer and V´eron gave in 2001 sharp conditions which imply that any solution of in Ω × (0, ∞) ∂t u − ∆u + b(x)|u|q−1 u = 0 ∂n u = 0 in ∂Ω × (0, ∞) u(x, 0) = u0 (x) in Ω becomes identically zero for t large enough. The method is associated to semiclassical limit of some Shr¨ odinger operator. They were initiated in a paper by Kondratiev and V´eron in 1997. The subject of the stage is to read the article in which they study the problem and the counter examples proving the sharpness of their result. Sharp extensions are also given by Belaud and Shishkov. References [1]Y Belaud, B. Helffer & L. V´eron : Long time vanishing property of solutions of some semilinear parabolic equations, Ann. I.H.P. Anal. Nonlin´ eaire 18, 43-68 (2001). [2]V.A. Kondratiev & L. V´eronAsymptotic behaviour of solutions of some nonlinear parabolic or elliptic equations, Asymptotic Analysis 14, 117-156 (1997). [3]Y. Belaud, Y. & A. Shishkov Long-time extinction of solutions of some semilinear parabolic equations, J. Differential Equations 238, 64-86(2007).
1. Yves BELAUD, LMPT, UMR 7350, Facult´e des Sciences et Techniques, Parc de Grandmont, 37200 Tours.
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