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Non-unicité des solutions d’une équation d’évolution non- linéaire. (French) Zbl 0553.35046
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
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References:
[1] P. Baras . Compacité de l’opérateur f \rightarrow u solution d’une équation non linéaire (du/dt) + Au \ni f . Note au C.R.A.S. , t. 286 , série A , ( 1978 ), p. 1113 - 1116 . MR 493554 | Zbl 0389.47030 · Zbl 0389.47030
[2] A. Haraux , F.B. Weissler . Non-uniqueness for a semi-linear initial value problem . Indiana University Mathematics Journal , vol. 31 , n^\circ 2 , ( 1982 ), p. 167 - 189 . MR 648169 | Zbl 0465.35049 · Zbl 0465.35049 · doi:10.1512/iumj.1982.31.31016
[3] A. Haraux , M. Kirane . Estimations C1 pour des problèmes paraboliques semi-linéaires . Université Pierre et Marie Curie . Paris V . · Zbl 0531.35048
[4] F.B. Weissler . Local existence and non-existence for semilinear parabolic equations in Lp . Indiana University Mathematics Journal , vol. 29 , n^\circ 1 , ( 1980 ), p. 79 - 102 . MR 554819 | Zbl 0443.35034 · Zbl 0443.35034 · doi:10.1512/iumj.1980.29.29007
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