Spectral properties of stationary solutions of the nonlinear heat equation
| Autor: | Cazenave, Thierry, Dickstein, Flavio, Weissler, Fred B., Centre national de la recherche scientifique (França). Laboratoire Jacques-Louis Lions |
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| Přispěvatelé: | Laboratoire Jacques-Louis Lions (LJLL), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Universidade Federal do Rio de Janeiro (UFRJ), Laboratoire Analyse, Géométrie et Applications (LAGA), Université Paris 8 Vincennes-Saint-Denis (UP8)-Centre National de la Recherche Scientifique (CNRS)-Université Sorbonne Paris Nord |
| Jazyk: | angličtina |
| Rok vydání: | 2011 |
| Předmět: |
Unit sphere
General Mathematics 35B35 01 natural sciences Finite-time blowup symbols.namesake Semilinear heat equation linearized operator Initial value problem [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] 0101 mathematics ComputingMilieux_MISCELLANEOUS Mathematics Linearized operator Operator (physics) 010102 general mathematics Mathematical analysis Spectral properties Sign-changing stationary finite-time blowup Eigenfunction 35J60 010101 applied mathematics Solutions Nonlinear heat equation Dirichlet boundary condition Product (mathematics) 35K55 symbols sign-changing stationary solutions |
| Zdroj: | Publicacions Matemàtiques Publicacions Matemàtiques, 2011, 55, pp.185-200. ⟨10.5565/publmat_55111_09⟩ Publicacions Matemàtiques; Vol. 55, Núm. 1 (2011); p. 185-200 Recercat. Dipósit de la Recerca de Catalunya instname Publ. Mat. 55, no. 1 (2011), 185-200 Recercat: Dipósit de la Recerca de Catalunya Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya) Dipòsit Digital de Documents de la UAB Universitat Autònoma de Barcelona |
| DOI: | 10.5565/publmat_55111_09⟩ |
| Popis: | In this paper, we prove that if $\Psi $ is a radially symmetric, sign-changing stationary solution of the nonlinear heat equation \begin{equation} \label{fAbs} u_t -\Delta u= |u|^\alpha u, \tag{NLH} \end{equation} ¶ in the unit ball of ${\mathbb R}^N $, $N= 3$, with Dirichlet boundary conditions, then the solution of \eqref{fAbs} with initial value $\lambda \Psi$ blows up in finite time if $|\lambda -1|>0$ is sufficiently small and if $\alpha >0$ is sufficiently small. The proof depends on showing that the inner product of $\Psi $ with the first eigenfunction of the linearized operator $L= -\Delta - (\alpha +1) | \Psi | ^\alpha $ is nonzero. |
| Databáze: | OpenAIRE |
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