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Control of weakly blowing up semilinear heat equations

 

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dc.contributor.editor Berestycki, Henri es
dc.contributor.editor Pomeau, Yves es
dc.creator Fernández Cara, Enrique es
dc.creator Zuazua Iriondo, Enrique es
dc.date.accessioned 2017-03-02T13:15:08Z
dc.date.available 2017-03-02T13:15:08Z
dc.date.issued 2002
dc.identifier.isbn 9781402009730 es
dc.identifier.isbn 9789401003070 es
dc.identifier.issn 1389-2185 es
dc.identifier.uri http://hdl.handle.net/11441/55140
dc.description.abstract In these notes we consider a semilinear heat equation in a bounded domain of IRd , with control on a subdomain and homogeneous Dirichlet boundary conditions. We consider nonlinearities for which, in the absence of control, blow up arises. We prove that when the nonlinearity grows at infinity fast enough, due to the local (in space) nature of the blow up phenomena, the control may not avoid the blow up to occur for suitable initial data. This is done by means of localized energy estimates. However, we also show that when the nonlinearity is weak enough, and provided the system admits a globally defined solution (for some initial data and control), the choice of a suitable control guarantees the global existence of solutions and moreover that the solution may be driven in any finite time to the globally defined solution. In order for this to be true we require the nonlinearity f to satisfy at infinity the growth condition f(s) |s| log3/2 (1 + |s|) → 0 as |s| → ∞. This is done by means of a fixed point argument and a careful analysis of the control of linearized heat equations relying on global Carleman estimates. The problem of controlling the blow up in this sense remains open for nonlinearities growing at infinity like f(s) ∼ |s|logp (1 + |s|) with 3/2 ≤ p ≤ 2. es
dc.format application/pdf es
dc.language.iso eng es
dc.publisher Springer es
dc.relation.ispartof Nonlinear PDE’s in condensed matter and reactive flows es
dc.rights Attribution-NonCommercial-NoDerivatives 4.0 Internacional *
dc.rights.uri http://creativecommons.org/licenses/by-nc-nd/4.0/ *
dc.title Control of weakly blowing up semilinear heat equations es
dc.type info:eu-repo/semantics/bookPart es
dc.type.version info:eu-repo/semantics/submittedVersion es
dc.rights.accessrights info:eu-repo/semantics/openAccess es
dc.contributor.affiliation Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico es
dc.relation.projectID PB95-1242 es
dc.relation.projectID PB96-0663 es
dc.relation.publisherversion https://link.springer.com/chapter/10.1007/978-94-010-0307-0_6 es
dc.identifier.doi 10.1007/978-94-010-0307-0_6 es
dc.contributor.group Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software es
idus.format.extent 21 p. es
dc.publication.initialPage 127 es
dc.publication.endPage 148 es
dc.relation.publicationplace Dordrecht es
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