On non-denseness for a method of fundamental solutions with source points fixed in time for parabolic equations

Johansson, B. Tomas

Comptes Rendus. Mathématique, Tome 359 (2021) no. 6, pp. 733-738.

Des combinaisons linéaires de solutions fondamentales avec des points sources indépendants du temps pour la résolution de l’équation de la chaleur sont étudiées. On étudie la question ouverte de savoir si ces combinaisons linéaires génèrent un ensemble dense dans l’espace des fonctions de carrés intégrables sur la limite latérale d’un cylindre espace-temps et on montre que la réponse à cette question est négative. L’indépendance linéaire de l’ensemble des solutions fondamentales est démontrée. Il est souligné à la fin pour un cas particulier que de telles combinaisons linéaires sont linéairement indépendantes et denses dans l’espace des fonctions de carrés intégrables définies sur la partie supérieure (où le temps est fixe) de la limite du cylindre espace-temps.

Linear combinations of fundamental solutions to the parabolic heat equation with source points fixed in time is investigated. The open problem whether these linear combinations generate a dense set in the space of square integrable functions on the lateral boundary of a space-time cylinder, is settled in the negative. Linear independence of the set of fundamental solutions is shown to hold. It is outlined at the end, for a particular example, that such linear combinations constitute a linearly independent and dense set in the space of square integrable functions on the upper top part (where time is fixed) of the boundary of this space-time cylinder.

Classification : 35K05, 35A08, 65N80

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