Asymptotics for a free boundary model in price formation
We study the asymptotics for a large time of solutions to a one-dimensional parabolic evolution equation with non-standard measure-valued right hand side, that involves derivatives of the solution computed at a free boundary point. The problem is a particular case of a mean-field free boundary model...
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| Veröffentlicht in: | Nonlinear analysis 2011-07, Vol.74 (10), p.3269-3294 |
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| container_title | Nonlinear analysis |
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| creator | González, María del Mar Gualdani, Maria Pia |
| description | We study the asymptotics for a large time of solutions to a one-dimensional parabolic evolution equation with non-standard measure-valued right hand side, that involves derivatives of the solution computed at a free boundary point. The problem is a particular case of a mean-field free boundary model proposed by Lasry–Lions on price formation and dynamic equilibria.
The main step in the proof is based on the fact that the free boundary disappears in the linearized problem, thus it can be treated as a perturbation through semigroup theory. This requires a delicate choice for the function spaces since higher regularity is needed near the free boundary. We show global existence for solutions with initial data in a small neighborhood of any equilibrium point, and exponential decay towards a stationary state. Moreover, the family of equilibria of the equation is stable, as follows from center manifold theory. |
| doi_str_mv | 10.1016/j.na.2011.02.005 |
| format | Article |
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The main step in the proof is based on the fact that the free boundary disappears in the linearized problem, thus it can be treated as a perturbation through semigroup theory. This requires a delicate choice for the function spaces since higher regularity is needed near the free boundary. We show global existence for solutions with initial data in a small neighborhood of any equilibrium point, and exponential decay towards a stationary state. Moreover, the family of equilibria of the equation is stable, as follows from center manifold theory.</description><identifier>ISSN: 0362-546X</identifier><identifier>EISSN: 1873-5215</identifier><identifier>DOI: 10.1016/j.na.2011.02.005</identifier><identifier>CODEN: NOANDD</identifier><language>eng</language><publisher>Amsterdam: Elsevier Ltd</publisher><subject>Asymptotic properties ; Asymptotics ; Center manifold ; Decay ; Derivatives ; Evolution ; Exact sciences and technology ; Free boundaries ; Free boundary ; Global analysis, analysis on manifolds ; Group theory ; Group theory and generalizations ; Mathematical analysis ; Mathematical models ; Mathematics ; Partial differential equations ; Perturbation methods ; Price formation ; Reaction–diffusion ; Sciences and techniques of general use ; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds</subject><ispartof>Nonlinear analysis, 2011-07, Vol.74 (10), p.3269-3294</ispartof><rights>2011 Elsevier Ltd</rights><rights>2015 INIST-CNRS</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c398t-54a21460f56caba23e78b1571a019288809d7bec771f2683d2e5ebb5473aa8f63</citedby><cites>FETCH-LOGICAL-c398t-54a21460f56caba23e78b1571a019288809d7bec771f2683d2e5ebb5473aa8f63</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktohtml>$$Uhttps://dx.doi.org/10.1016/j.na.2011.02.005$$EHTML$$P50$$Gelsevier$$H</linktohtml><link.rule.ids>314,780,784,3550,27924,27925,45995</link.rule.ids><backlink>$$Uhttp://pascal-francis.inist.fr/vibad/index.php?action=getRecordDetail&idt=24103354$$DView record in Pascal Francis$$Hfree_for_read</backlink></links><search><creatorcontrib>González, María del Mar</creatorcontrib><creatorcontrib>Gualdani, Maria Pia</creatorcontrib><title>Asymptotics for a free boundary model in price formation</title><title>Nonlinear analysis</title><description>We study the asymptotics for a large time of solutions to a one-dimensional parabolic evolution equation with non-standard measure-valued right hand side, that involves derivatives of the solution computed at a free boundary point. The problem is a particular case of a mean-field free boundary model proposed by Lasry–Lions on price formation and dynamic equilibria.
The main step in the proof is based on the fact that the free boundary disappears in the linearized problem, thus it can be treated as a perturbation through semigroup theory. This requires a delicate choice for the function spaces since higher regularity is needed near the free boundary. We show global existence for solutions with initial data in a small neighborhood of any equilibrium point, and exponential decay towards a stationary state. Moreover, the family of equilibria of the equation is stable, as follows from center manifold theory.</description><subject>Asymptotic properties</subject><subject>Asymptotics</subject><subject>Center manifold</subject><subject>Decay</subject><subject>Derivatives</subject><subject>Evolution</subject><subject>Exact sciences and technology</subject><subject>Free boundaries</subject><subject>Free boundary</subject><subject>Global analysis, analysis on manifolds</subject><subject>Group theory</subject><subject>Group theory and generalizations</subject><subject>Mathematical analysis</subject><subject>Mathematical models</subject><subject>Mathematics</subject><subject>Partial differential equations</subject><subject>Perturbation methods</subject><subject>Price formation</subject><subject>Reaction–diffusion</subject><subject>Sciences and techniques of general use</subject><subject>Topology. Manifolds and cell complexes. 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Global analysis and analysis on manifolds</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>González, María del Mar</creatorcontrib><creatorcontrib>Gualdani, Maria Pia</creatorcontrib><collection>Pascal-Francis</collection><collection>CrossRef</collection><collection>Computer and Information Systems Abstracts</collection><collection>Mechanical & Transportation Engineering Abstracts</collection><collection>Technology Research Database</collection><collection>Engineering Research Database</collection><collection>ProQuest Computer Science Collection</collection><collection>Civil Engineering Abstracts</collection><collection>Advanced Technologies Database with Aerospace</collection><collection>Computer and Information Systems Abstracts Academic</collection><collection>Computer and Information Systems Abstracts Professional</collection><jtitle>Nonlinear analysis</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>González, María del Mar</au><au>Gualdani, Maria Pia</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Asymptotics for a free boundary model in price formation</atitle><jtitle>Nonlinear analysis</jtitle><date>2011-07-01</date><risdate>2011</risdate><volume>74</volume><issue>10</issue><spage>3269</spage><epage>3294</epage><pages>3269-3294</pages><issn>0362-546X</issn><eissn>1873-5215</eissn><coden>NOANDD</coden><abstract>We study the asymptotics for a large time of solutions to a one-dimensional parabolic evolution equation with non-standard measure-valued right hand side, that involves derivatives of the solution computed at a free boundary point. The problem is a particular case of a mean-field free boundary model proposed by Lasry–Lions on price formation and dynamic equilibria.
The main step in the proof is based on the fact that the free boundary disappears in the linearized problem, thus it can be treated as a perturbation through semigroup theory. This requires a delicate choice for the function spaces since higher regularity is needed near the free boundary. We show global existence for solutions with initial data in a small neighborhood of any equilibrium point, and exponential decay towards a stationary state. Moreover, the family of equilibria of the equation is stable, as follows from center manifold theory.</abstract><cop>Amsterdam</cop><pub>Elsevier Ltd</pub><doi>10.1016/j.na.2011.02.005</doi><tpages>26</tpages><oa>free_for_read</oa></addata></record> |
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| subjects | Asymptotic properties Asymptotics Center manifold Decay Derivatives Evolution Exact sciences and technology Free boundaries Free boundary Global analysis, analysis on manifolds Group theory Group theory and generalizations Mathematical analysis Mathematical models Mathematics Partial differential equations Perturbation methods Price formation Reaction–diffusion Sciences and techniques of general use Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds |
| title | Asymptotics for a free boundary model in price formation |
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