1 Universit´ e Fran¸ cois Rabelais 2010-2011 Laboratoire de Math´ ematiques et Physique Th´ eorique CNRS UMR 6083 F´ ed´ eration Denis-Poisson Master thesis proposal Y. Belaud and L. V´ eron [email protected] [email protected] Title : Heat equation with singular potential Consider the problem ∂t u − ∆u = V (x)u + f (x, t) u(x, 0) = u0 (x)

in RN × (0, ∞) in RN

(0.1)

where V is a positive singular potential. The starting observation is a classical result by Baras and Goldstein dealing with the case V (x) = c|x|−2 : if un is the solution of the regularized problem ∂t u − ∆u = Vn (x)u u(x, 0) = u0 (x)

in RN × (0, ∞) in RN

(0.2)

∗ := (N − 2)2 /4 implies that u converges to where Vn (x) = min{n, c|x|−2 }, then c ≤ CN n ∗ , then u (x, t) ↑ ∞ for every a solution of (0.1 ) defined on RN × (0, ∞), while if c > CN n t > 0. Two proofs are given, one of them probabilistic. An interesting extension is given by Cabre and Martel which allowed to give almost necessary and sufficient conditions for the intantaneous blow-up of solutions. With this extension Goldstein and Zhang were able to replace the Laplacian by a symmetric operator u 7→ div (A∇u). This extension is given in the article [4].

The master thesis will consist in reading and summarizing the papers [1]–[4] (except may be the probabilistic part in [1], if the student is not acquainted with probability theory). The associated problem with a negative singular potential is essentially open. The method of truncating the potential and the detruncating it leads to a decreasing sequence of approximate solutions. Underwhat conditions does it converge to 0 ? In that case the potential could also depend on the t variable with blow-up at t = 0 ; another deep associated question is the existence of the initial trace of positive solutions of ∂t u − ∆u + V (x, t)u = 0

in RN × (0, ∞).

(0.3)

Many applications to semilinear problems could follow. References [1] Baras, P. and Goldstein, J. A. The heat equation with a singular potential, Trans. Amer. Math. Soc., 284 (1984), 121-139. The starting article

2 [2] Baras, P. and Goldstein, J. A., Remarks on the inverse square potential in quantum mechanics, North-Holland Math. Stud., 92 (1984), 31-35. [3] Cabre X. and Martel Y., Existence versus explosion instantanee pour des equations de la chaleur linaires avec potentiel singulier, C. R. Acad. Sci. Paris, t. 329, S´erie I, p. 973-978, 1999. A nice extension Further studies [4] Goldstein, J. A. and Qi S. Zhang, Linear parabolic equations with strong singular potentials, Trans. Amer. Math. Soc., 355 (2002), 197-211.