Time-dependent singularities in semilinear parabolic equations: Behavior at the singularities

详细信息    查看全文

• 作者：Toru Kan ; ^{kan@math.titech.ac.jp" class="auth_mail" title="E-mail the corresponding author} ; Jin Takahashi ^{takahashi.j.ab@m.titech.ac.jp" class="auth_mail" title="E-mail the corresponding author}
• 关键词：35K58 ; 35A20 ; 35B60
• 刊名：Journal of Differential Equations
• 出版年：2016
• 出版时间：15 May 2016
• 年：2016
• 卷：260
• 期：10
• 页码：7278-7319
• 全文大小：482 K

文摘

Singularities of solutions of semilinear parabolic equations are discussed. A typical equation is class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616000358&_mathId=si1.gif&_user=111111111&_pii=S0022039616000358&_rdoc=1&_issn=00220396&md5=e76006fd0b1e799a41738ed6ccc0c867" title="Click to view the MathML source">∂_tu−Δu=u^pclass="mathContainer hidden">class="mathCode">${\partial }_{t}u-\mathrm{\Delta }u=u^p$, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616000358&_mathId=si2.gif&_user=111111111&_pii=S0022039616000358&_rdoc=1&_issn=00220396&md5=5a6b54b8de68f0e0e36a246299ea83d5" title="Click to view the MathML source">x∈R^N∖{ξ(t)}class="mathContainer hidden">class="mathCode">$x\in {\mathbb{R}}^N\setminus \{\xi (t)\}$, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616000358&_mathId=si3.gif&_user=111111111&_pii=S0022039616000358&_rdoc=1&_issn=00220396&md5=73644b4213ee593de72cd8d7ef35c655" title="Click to view the MathML source">t∈Iclass="mathContainer hidden">class="mathCode">$t\in I$. Here class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616000358&_mathId=si4.gif&_user=111111111&_pii=S0022039616000358&_rdoc=1&_issn=00220396&md5=526631b0dd8797115b74336af72231f8" title="Click to view the MathML source">N≥2class="mathContainer hidden">class="mathCode">$N\ge 2$, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616000358&_mathId=si5.gif&_user=111111111&_pii=S0022039616000358&_rdoc=1&_issn=00220396&md5=3340243036bb554f6652f5f58822768a" title="Click to view the MathML source">p>1class="mathContainer hidden">class="mathCode">$p>1$, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616000358&_mathId=si6.gif&_user=111111111&_pii=S0022039616000358&_rdoc=1&_issn=00220396&md5=53df8bf81fd18fd3550047145cfa801c" title="Click to view the MathML source">I⊂Rclass="mathContainer hidden">class="mathCode">$I\subset \mathbb{R}$ is an open interval and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616000358&_mathId=si7.gif&_user=111111111&_pii=S0022039616000358&_rdoc=1&_issn=00220396&md5=23bfdbb4905768c05c9aaee29a392df0" title="Click to view the MathML source">ξ∈C^α(I;R^N)class="mathContainer hidden">class="mathCode">$\xi \in C^{\alpha }(I;{\mathbb{R}}^N)$ with class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616000358&_mathId=si8.gif&_user=111111111&_pii=S0022039616000358&_rdoc=1&_issn=00220396&md5=0dcc16de6fbb3450b42e1f180cf2eacd" title="Click to view the MathML source">α>1/2class="mathContainer hidden">class="mathCode">$\alpha >1/2$. For this equation it is shown that every nonnegative solution u   satisfies class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616000358&_mathId=si9.gif&_user=111111111&_pii=S0022039616000358&_rdoc=1&_issn=00220396&md5=14ba12ab96fc226b0151902bb34948fc" title="Click to view the MathML source">∂_tu−Δu=u^p+Λclass="mathContainer hidden">class="mathCode">${\partial }_{t}u-\mathrm{\Delta }u=u^p+\mathrm{\Lambda }$ in class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616000358&_mathId=si10.gif&_user=111111111&_pii=S0022039616000358&_rdoc=1&_issn=00220396&md5=eec253e49dd5b4b93acf60f67c1f2bef" title="Click to view the MathML source">D^′(R^N×I)class="mathContainer hidden">class="mathCode">${\mathcal{D}}^{\prime }({\mathbb{R}}^N\times I)$ for some measure Λ whose support is contained in class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616000358&_mathId=si11.gif&_user=111111111&_pii=S0022039616000358&_rdoc=1&_issn=00220396&md5=ccdeac3538474703c315c5891d23ee74" title="Click to view the MathML source">{(ξ(t),t);t∈I}class="mathContainer hidden">class="mathCode">$\{(\xi (t),t);t\in I\}$. Moreover, if class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616000358&_mathId=si12.gif&_user=111111111&_pii=S0022039616000358&_rdoc=1&_issn=00220396&md5=d61e3e11b9d978b6efb519e4782ea440" title="Click to view the MathML source">(N−2)p<Nclass="mathContainer hidden">class="mathCode">$(N-2)p<N$, then class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616000358&_mathId=si13.gif&_user=111111111&_pii=S0022039616000358&_rdoc=1&_issn=00220396&md5=a5bd74186a3bc8954a58acbdf2b6b2d5" title="Click to view the MathML source">u(x,t)=(a(t)+o(1))Ψ(x−ξ(t))class="mathContainer hidden">class="mathCode">$u(x,t)=(a(t)+o(1))\mathrm{\Psi }(x-\xi (t))$ for almost every class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616000358&_mathId=si3.gif&_user=111111111&_pii=S0022039616000358&_rdoc=1&_issn=00220396&md5=73644b4213ee593de72cd8d7ef35c655" title="Click to view the MathML source">t∈Iclass="mathContainer hidden">class="mathCode">$t\in I$ as class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616000358&_mathId=si14.gif&_user=111111111&_pii=S0022039616000358&_rdoc=1&_issn=00220396&md5=e327e46ffbf110f0625cf5ff712cd6a7" title="Click to view the MathML source">x→ξ(t)class="mathContainer hidden">class="mathCode">$x\to \xi (t)$, where Ψ is the fundamental solution of Laplace's equation in class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616000358&_mathId=si15.gif&_user=111111111&_pii=S0022039616000358&_rdoc=1&_issn=00220396&md5=11a02bef2091a0af4f78eb4f7ab7b31b" title="Click to view the MathML source">R^Nclass="mathContainer hidden">class="mathCode">${\mathbb{R}}^N$ and a is some function determined by Λ.