A complete characterisation of local existence for semilinear heat equations in Lebesgue spaces

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• 作者：R. Laisterp>ap>p> ; p>p>Robert.Laister@uwe.ac.uk" class="auth_mail" title="E-mail the corresponding authorp> ; J.C. Robinsonp>bp>p> ; p>p>J.C.Robinson@warwick.ac.uk" class="auth_mail" title="E-mail the corresponding authorp> ; M. Sierżęgap>bp>p> ; p>p>M.L.Sierzega@warwick.ac.uk" class="auth_mail" title="E-mail the corresponding authorp> ; A. Vidal-Ló ; pezp>cp>p> ; p>p>Alejandro.Vidal@xjtlu.edu.cn" class="auth_mail" title="E-mail the corresponding authorp>
• 关键词：Semilinear heat equation ; Dirichlet problem ; Local existence ; Non-existence ; Instantaneous blow-up ; Dirichlet heat kernel
• 刊名：Annales de l'Institut Henri Poincare (C) Non Linear Analysis
• 出版年：2016
• 出版时间：November-December 2016
• 年：2016
• 卷：33
• 期：6
• 页码：1519-1538
• 全文大小：368 K

文摘

We consider the scalar semilinear heat equation pan id="mmlsi1" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0294144915000645&_mathId=si1.gif&_user=111111111&_pii=S0294144915000645&_rdoc=1&_issn=02941449&md5=d926f4996fd7581c8453b7cc967906a2" title="Click to view the MathML source">u_t−Δu=f(u)pan>pan class="mathContainer hidden">pan class="mathCode">pan>pan>pan>, where pan id="mmlsi2" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0294144915000645&_mathId=si2.gif&_user=111111111&_pii=S0294144915000645&_rdoc=1&_issn=02941449&md5=1233ea1621cfd5133fda5e7ba0ffa9f0" title="Click to view the MathML source">f:[0,∞)→[0,∞)pan>pan class="mathContainer hidden">pan class="mathCode">$f:[0,\infty )\to [0,\infty )$pan>pan>pan> is continuous and non-decreasing but need not be convex. We completely characterise those functions f   for which the equation has a local solution bounded in pan id="mmlsi133" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0294144915000645&_mathId=si133.gif&_user=111111111&_pii=S0294144915000645&_rdoc=1&_issn=02941449&md5=80c94c03982c570f1736f573a464954b" title="Click to view the MathML source">Lp>qp>(Ω)pan>pan class="mathContainer hidden">pan class="mathCode">pan>pan>pan> for all non-negative initial data pan id="mmlsi100" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0294144915000645&_mathId=si100.gif&_user=111111111&_pii=S0294144915000645&_rdoc=1&_issn=02941449&md5=9f6a9a8667650346fd2f46b481cc98a5" title="Click to view the MathML source">u₀∈Lp>qp>(Ω)pan>pan class="mathContainer hidden">pan class="mathCode">pan>pan>pan>, when pan id="mmlsi5" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0294144915000645&_mathId=si5.gif&_user=111111111&_pii=S0294144915000645&_rdoc=1&_issn=02941449&md5=c0c809c2bc06fa153ad639a44af453cf" title="Click to view the MathML source">Ω⊂Rp>dp>pan>pan class="mathContainer hidden">pan class="mathCode">pan>pan>pan> is a bounded domain with Dirichlet boundary conditions. For pan id="mmlsi132" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0294144915000645&_mathId=si132.gif&_user=111111111&_pii=S0294144915000645&_rdoc=1&_issn=02941449&md5=d15acf7ad379f2b0a67221326e4a738d" title="Click to view the MathML source">q∈(1,∞)pan>pan class="mathContainer hidden">pan class="mathCode">$q\in (1,\infty )$pan>pan>pan> this holds if and only if pan id="mmlsi7" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0294144915000645&_mathId=si7.gif&_user=111111111&_pii=S0294144915000645&_rdoc=1&_issn=02941449&md5=61a3be373e59d24704acdc9729e4cc38">dth="216" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0294144915000645-si7.gif">pt>der="0" style="vertical-align:bottom" width="216" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0294144915000645-si7.gif">pt>pan class="mathContainer hidden">pan class="mathCode">dth="0.2em">pace>mal">sups→∞pace width="0.2em">pace>p>s−(1+2q/d)p>f(s)<∞pan>pan>pan>; and for pan id="mmlsi8" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0294144915000645&_mathId=si8.gif&_user=111111111&_pii=S0294144915000645&_rdoc=1&_issn=02941449&md5=68204ad5e8c0fe5c9a486be277588179" title="Click to view the MathML source">q=1pan>pan class="mathContainer hidden">pan class="mathCode">$q=1$pan>pan>pan> if and only if pan id="mmlsi9" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0294144915000645&_mathId=si9.gif&_user=111111111&_pii=S0294144915000645&_rdoc=1&_issn=02941449&md5=73683f702a3e792c5a0644113d99d0cf">dth="172" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0294144915000645-si9.gif">pt>der="0" style="vertical-align:bottom" width="172" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0294144915000645-si9.gif">pt>pan class="mathContainer hidden">pan class="mathCode">dth="0.2em">pace>mal">ds<∞pan>pan>pan>, where pan id="mmlsi10" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0294144915000645&_mathId=si10.gif&_user=111111111&_pii=S0294144915000645&_rdoc=1&_issn=02941449&md5=5aed9aca5ebed42dfad174dc994c6313" title="Click to view the MathML source">F(s)=sup_1≤t≤s⁡f(t)/tpan>pan class="mathContainer hidden">pan class="mathCode">pan>pan>pan>. This shows for the first time that the model nonlinearity pan id="mmlsi11" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0294144915000645&_mathId=si11.gif&_user=111111111&_pii=S0294144915000645&_rdoc=1&_issn=02941449&md5=767b4b27949104aede1c64bd4ac4642a" title="Click to view the MathML source">f(u)=up>1+2q/dp>pan>pan class="mathContainer hidden">pan class="mathCode">pan>pan>pan> is truly the ‘boundary case’ when pan id="mmlsi132" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0294144915000645&_mathId=si132.gif&_user=111111111&_pii=S0294144915000645&_rdoc=1&_issn=02941449&md5=d15acf7ad379f2b0a67221326e4a738d" title="Click to view the MathML source">q∈(1,∞)pan>pan class="mathContainer hidden">pan class="mathCode">$q\in (1,\infty )$pan>pan>pan>, but that this is not true for pan id="mmlsi8" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0294144915000645&_mathId=si8.gif&_user=111111111&_pii=S0294144915000645&_rdoc=1&_issn=02941449&md5=68204ad5e8c0fe5c9a486be277588179" title="Click to view the MathML source">q=1pan>pan class="mathContainer hidden">pan class="mathCode">$q=1$pan>pan>pan>.p><p id="sp0020">The same characterisations hold for the equation posed on the whole space pan id="mmlsi12" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0294144915000645&_mathId=si12.gif&_user=111111111&_pii=S0294144915000645&_rdoc=1&_issn=02941449&md5=c7bb1a6224c553c1825787197583dfcd" title="Click to view the MathML source">Rp>dp>pan>pan class="mathContainer hidden">pan class="mathCode">pan>pan>pan> provided that pan id="mmlsi13" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0294144915000645&_mathId=si13.gif&_user=111111111&_pii=S0294144915000645&_rdoc=1&_issn=02941449&md5=cb8047436a05ef97c06f6069904c4955">dth="157" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0294144915000645-si13.gif">pt>der="0" style="vertical-align:bottom" width="157" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0294144915000645-si13.gif">pt>pan class="mathContainer hidden">pan class="mathCode">dth="0.2em">pace>mal">sups→0pace width="0.2em">pace>f(s)/s<∞pan>pan>pan>.