Non-unicité des solutions d’une équation d’évolution non- linéaire. (French) Zbl 0553.35046

Baras, Pierre

Ann. Fac. Sci. Toulouse, V. Sér., Math. 5, 287-302 (1983).

Author’s summary: We prove a non-uniqueness result for a semi-linear parabolic equation. If \(\gamma >1\) and N an integer such that \(1<N(\gamma -1)/2<\inf (\gamma,(\gamma +3)/2))\) and \(\Omega\) is a ball of \({\mathbb{R}}^ N\), then the equation \(\partial u/\partial t-\Delta u=u| u|^{\gamma -1},\quad u(0)=u_ 0,\quad u|_{\partial \Omega}=0\) possesses an infinite number of solutions such that \(\lim_{t\to 0}\| u(t)-u_ 0\|_ q=0\) where \(1\leq q<N(\gamma - 1)/2\).

Reviewer: R.Badescu

MSC:

35K60	Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35A05	General existence and uniqueness theorems (PDE) (MSC2000)
35B30	Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs

Keywords:

non-uniqueness; semi-linear parabolic equation; infinite number of solutions

References:

[1]	Baras, P.. «Compacité de l’opérateur f → u solution d’une équation non linéaire (du/dt) + Au ∋ f». Note au C.R.A.S., t. 286, série A, (1978), p. 1113-1116. · Zbl 0389.47030
[2]	Haraux, A., Weissler, F.B.. «Non-uniqueness for a semi-linear initial value problem». Indiana University Mathematics Journal, vol. 31, n° 2, (1982), p. 167-189. · Zbl 0465.35049
[3]	Haraux, A., Kirane, M.. «Estimations C1 pour des problèmes paraboliques semi-linéaires». Université Pierre et Marie Curie. ParisV. · Zbl 0494.35054
[4]	Weissler, F.B.. «Local existence and non-existence for semilinear parabolic equations in Lp». Indiana University Mathematics Journal, vol. 29, n° 1, (1980), p. 79-102. · Zbl 0443.35034