THE BLOW-UP PROFILE FOR A FAST DIFFUSION EQUATION WITH A NONLINEAR BOUNDARY CONDITION
We study positive solutions of a fast diffusion equation in the half-line with a nonlinear boundary condition, $\left\{ {\begin{array}{*{20}{c}} {{u_t} = {{\left( {{u^m}}\right)}_{xx}}\,\left( {x,t} \right) \in {R_ + } \times \left( {0,T}\right),} \\ { - {{\left( {{u^m}} \right)}_x}\left( {0,t} \rig...
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| Veröffentlicht in: | The Rocky Mountain journal of mathematics 2003-03, Vol.33 (1), p.123-146 |
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| container_title | The Rocky Mountain journal of mathematics |
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| creator | FERREIRA, RAÚL DE PABLO, ARTURO QUIRÓS, FERNANDO ROSSI, JULIO D. |
| description | We study positive solutions of a fast diffusion equation in the half-line with a nonlinear boundary condition, $\left\{ {\begin{array}{*{20}{c}} {{u_t} = {{\left( {{u^m}}\right)}_{xx}}\,\left( {x,t} \right) \in {R_ + } \times \left( {0,T}\right),} \\ { - {{\left( {{u^m}} \right)}_x}\left( {0,t} \right) = {u^p}\left( {0,t} \right)\,t \in \left( {0,T} \right),} \\ {u\left( {x,0} \right) = {u_0}\left( x \right)\,x \in {R_{ + ,}}} \\ \end{array} } \right.$ where 0 < m < 1 and p > 0 are parameters. We describe in terms of p and m when all solutions exist globally in time, when all solutions blow up in a finite time, and when there are both blowing up and global solutions. For blowing up solutions we find the blow-up rate and the blow-up set and we describe the asymptotic behavior close to the blow-up time T in terms of a self-similar profile. |
| doi_str_mv | 10.1216/rmjm/1181069989 |
| format | Article |
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We describe in terms of p and m when all solutions exist globally in time, when all solutions blow up in a finite time, and when there are both blowing up and global solutions. For blowing up solutions we find the blow-up rate and the blow-up set and we describe the asymptotic behavior close to the blow-up time T in terms of a self-similar profile.</description><identifier>ISSN: 0035-7596</identifier><identifier>EISSN: 1945-3795</identifier><identifier>DOI: 10.1216/rmjm/1181069989</identifier><identifier>CODEN: RMJMAE</identifier><language>eng</language><publisher>Tempe, AZ: The Rocky Mountain Mathematics Consortium</publisher><subject>35B35 ; 35B40 ; 35K65 ; asymptotic behavior ; Blow-up ; Boundary conditions ; Critical points ; Differential equations ; Exact sciences and technology ; fast diffusion equation ; Heat equation ; Infinity ; Mathematical analysis ; Mathematical theorems ; Mathematics ; nonlinear boundary conditions ; Partial differential equations ; Porous materials ; Quadrants ; Sciences and techniques of general use ; Trajectories</subject><ispartof>The Rocky Mountain journal of mathematics, 2003-03, Vol.33 (1), p.123-146</ispartof><rights>Copyright © 2003 Rocky Mountain Mathematics Consortium</rights><rights>2004 INIST-CNRS</rights><rights>Copyright 2003 Rocky Mountain Mathematics Consortium</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c390t-f4210dbfdcd7fb8bbcf188eff5bf1d902e692614787ff861cfd7f44649a8e303</citedby><cites>FETCH-LOGICAL-c390t-f4210dbfdcd7fb8bbcf188eff5bf1d902e692614787ff861cfd7f44649a8e303</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://www.jstor.org/stable/pdf/44238915$$EPDF$$P50$$Gjstor$$H</linktopdf><linktohtml>$$Uhttps://www.jstor.org/stable/44238915$$EHTML$$P50$$Gjstor$$H</linktohtml><link.rule.ids>230,314,776,780,799,828,881,921,27901,27902,57992,57996,58225,58229</link.rule.ids><backlink>$$Uhttp://pascal-francis.inist.fr/vibad/index.php?action=getRecordDetail&idt=15073175$$DView record in Pascal Francis$$Hfree_for_read</backlink></links><search><creatorcontrib>FERREIRA, RAÚL</creatorcontrib><creatorcontrib>DE PABLO, ARTURO</creatorcontrib><creatorcontrib>QUIRÓS, FERNANDO</creatorcontrib><creatorcontrib>ROSSI, JULIO D.</creatorcontrib><title>THE BLOW-UP PROFILE FOR A FAST DIFFUSION EQUATION WITH A NONLINEAR BOUNDARY CONDITION</title><title>The Rocky Mountain journal of mathematics</title><description>We study positive solutions of a fast diffusion equation in the half-line with a nonlinear boundary condition, $\left\{ {\begin{array}{*{20}{c}} {{u_t} = {{\left( {{u^m}}\right)}_{xx}}\,\left( {x,t} \right) \in {R_ + } \times \left( {0,T}\right),} \\ { - {{\left( {{u^m}} \right)}_x}\left( {0,t} \right) = {u^p}\left( {0,t} \right)\,t \in \left( {0,T} \right),} \\ {u\left( {x,0} \right) = {u_0}\left( x \right)\,x \in {R_{ + ,}}} \\ \end{array} } \right.$ where 0 < m < 1 and p > 0 are parameters. We describe in terms of p and m when all solutions exist globally in time, when all solutions blow up in a finite time, and when there are both blowing up and global solutions. For blowing up solutions we find the blow-up rate and the blow-up set and we describe the asymptotic behavior close to the blow-up time T in terms of a self-similar profile.</description><subject>35B35</subject><subject>35B40</subject><subject>35K65</subject><subject>asymptotic behavior</subject><subject>Blow-up</subject><subject>Boundary conditions</subject><subject>Critical points</subject><subject>Differential equations</subject><subject>Exact sciences and technology</subject><subject>fast diffusion equation</subject><subject>Heat equation</subject><subject>Infinity</subject><subject>Mathematical analysis</subject><subject>Mathematical theorems</subject><subject>Mathematics</subject><subject>nonlinear boundary conditions</subject><subject>Partial differential equations</subject><subject>Porous materials</subject><subject>Quadrants</subject><subject>Sciences and techniques of general use</subject><subject>Trajectories</subject><issn>0035-7596</issn><issn>1945-3795</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2003</creationdate><recordtype>article</recordtype><recordid>eNptkMFrgzAUxsPYYF23806DXHZ0TUyiydG2ugpiOquUnUSjAaWdI3aH_fdTKu1lp_d43-_7HnwAPGP0hm3sLMyxPS4w5hg5QnBxA2ZYUGYRV7BbMEOIMMtlwrkHD33fIoQpE2QGsnTjw2Uk91a2hdtEBmHkw0Am0IOBt0vhOgyCbBfKGPofmZeOyz5MN4McyzgKY99L4FJm8dpLPuFKxutwZB7BnS4Off00zTlIAz9dbaxIvocrL7IUEehkaWpjVJW6UpWrS16WSmPOa61ZqXElkF07wnYwdbmrNXew0gNHqUNFwWuCyBx459hv07W1OtU_6tBU-bdpjoX5zbuiyVdZNF2nMdaUX2saMhbnDGW6vje1vtgxysdi_3G8Tl-LXhUHbYov1fRXG0MuwS4buJcz1_anzlx0Sm3CBWbkD4p4fPE</recordid><startdate>20030301</startdate><enddate>20030301</enddate><creator>FERREIRA, RAÚL</creator><creator>DE PABLO, ARTURO</creator><creator>QUIRÓS, FERNANDO</creator><creator>ROSSI, JULIO D.</creator><general>The Rocky Mountain Mathematics Consortium</general><general>Rocky Mountain Mathematics Consortium</general><scope>IQODW</scope><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>20030301</creationdate><title>THE BLOW-UP PROFILE FOR A FAST DIFFUSION EQUATION WITH A NONLINEAR BOUNDARY CONDITION</title><author>FERREIRA, RAÚL ; DE PABLO, ARTURO ; QUIRÓS, FERNANDO ; ROSSI, JULIO D.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c390t-f4210dbfdcd7fb8bbcf188eff5bf1d902e692614787ff861cfd7f44649a8e303</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2003</creationdate><topic>35B35</topic><topic>35B40</topic><topic>35K65</topic><topic>asymptotic behavior</topic><topic>Blow-up</topic><topic>Boundary conditions</topic><topic>Critical points</topic><topic>Differential equations</topic><topic>Exact sciences and technology</topic><topic>fast diffusion equation</topic><topic>Heat equation</topic><topic>Infinity</topic><topic>Mathematical analysis</topic><topic>Mathematical theorems</topic><topic>Mathematics</topic><topic>nonlinear boundary conditions</topic><topic>Partial differential equations</topic><topic>Porous materials</topic><topic>Quadrants</topic><topic>Sciences and techniques of general use</topic><topic>Trajectories</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>FERREIRA, RAÚL</creatorcontrib><creatorcontrib>DE PABLO, ARTURO</creatorcontrib><creatorcontrib>QUIRÓS, FERNANDO</creatorcontrib><creatorcontrib>ROSSI, JULIO D.</creatorcontrib><collection>Pascal-Francis</collection><collection>CrossRef</collection><jtitle>The Rocky Mountain journal of mathematics</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>FERREIRA, RAÚL</au><au>DE PABLO, ARTURO</au><au>QUIRÓS, FERNANDO</au><au>ROSSI, JULIO D.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>THE BLOW-UP PROFILE FOR A FAST DIFFUSION EQUATION WITH A NONLINEAR BOUNDARY CONDITION</atitle><jtitle>The Rocky Mountain journal of mathematics</jtitle><date>2003-03-01</date><risdate>2003</risdate><volume>33</volume><issue>1</issue><spage>123</spage><epage>146</epage><pages>123-146</pages><issn>0035-7596</issn><eissn>1945-3795</eissn><coden>RMJMAE</coden><abstract>We study positive solutions of a fast diffusion equation in the half-line with a nonlinear boundary condition, $\left\{ {\begin{array}{*{20}{c}} {{u_t} = {{\left( {{u^m}}\right)}_{xx}}\,\left( {x,t} \right) \in {R_ + } \times \left( {0,T}\right),} \\ { - {{\left( {{u^m}} \right)}_x}\left( {0,t} \right) = {u^p}\left( {0,t} \right)\,t \in \left( {0,T} \right),} \\ {u\left( {x,0} \right) = {u_0}\left( x \right)\,x \in {R_{ + ,}}} \\ \end{array} } \right.$ where 0 < m < 1 and p > 0 are parameters. We describe in terms of p and m when all solutions exist globally in time, when all solutions blow up in a finite time, and when there are both blowing up and global solutions. For blowing up solutions we find the blow-up rate and the blow-up set and we describe the asymptotic behavior close to the blow-up time T in terms of a self-similar profile.</abstract><cop>Tempe, AZ</cop><pub>The Rocky Mountain Mathematics Consortium</pub><doi>10.1216/rmjm/1181069989</doi><tpages>24</tpages><oa>free_for_read</oa></addata></record> |
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| source | Project Euclid; JSTOR; EZB Electronic Journals Library |
| subjects | 35B35 35B40 35K65 asymptotic behavior Blow-up Boundary conditions Critical points Differential equations Exact sciences and technology fast diffusion equation Heat equation Infinity Mathematical analysis Mathematical theorems Mathematics nonlinear boundary conditions Partial differential equations Porous materials Quadrants Sciences and techniques of general use Trajectories |
| title | THE BLOW-UP PROFILE FOR A FAST DIFFUSION EQUATION WITH A NONLINEAR BOUNDARY CONDITION |
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