Global existence and finite time blow-up of solutions of a Gierer-Meinhardt system

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• 作者：Fang Li^a ; ^{fangli0214@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; Rui Peng^b ; ^{pengrui_seu@163.com" class="auth_mail" title="E-mail the corresponding author} ; Xianfa Song^c ; ^{songxianfa@tju.edu.cn" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Gierer&ndash ; Meinhardt system ; Global existence ; Finite time blow-up
• 刊名：Journal of Differential Equations
• 出版年：2017
• 出版时间：5 January 2017
• 年：2017
• 卷：262
• 期：1
• 页码：559-589
• 全文大小：415 K

文摘

We are concerned with the Gierer–Meinhardt system with zero Neumann boundary condition: where class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616303138&_mathId=si2.gif&_user=111111111&_pii=S0022039616303138&_rdoc=1&_issn=00220396&md5=eccdbcedc1600e12c601cad7e75699b7" 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-S0022039616303138&_mathId=si3.gif&_user=111111111&_pii=S0022039616303138&_rdoc=1&_issn=00220396&md5=15b3183b91a40cf31a46eb53f25215ec" title="Click to view the MathML source">s>−1class="mathContainer hidden">class="mathCode">$s>-1$, class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616303138&_mathId=si4.gif&_user=111111111&_pii=S0022039616303138&_rdoc=1&_issn=00220396&md5=68954519f06eaa3bdcc7191080f4f240">class="imgLazyJSB inlineImage" height="15" width="125" alt="View the MathML source" style="margin-top: -5px; vertical-align: middle" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022039616303138-si4.gif">class="mathContainer hidden">class="mathCode">$q,r,d_1,d_2,a_1,a_2$ are positive constants, class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616303138&_mathId=si5.gif&_user=111111111&_pii=S0022039616303138&_rdoc=1&_issn=00220396&md5=786f5675b19b27d9c988e2f3e41f32f0">class="imgLazyJSB inlineImage" height="14" width="89" alt="View the MathML source" style="margin-top: -5px; vertical-align: middle" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022039616303138-si5.gif">class="mathContainer hidden">class="mathCode">${\delta }_1,{\delta }_2,u_0,v_0$ are nonnegative smooth functions, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616303138&_mathId=si6.gif&_user=111111111&_pii=S0022039616303138&_rdoc=1&_issn=00220396&md5=89ce441b683e055992000ceb4291830c" title="Click to view the MathML source">Ω⊂R^dclass="mathContainer hidden">class="mathCode">$\mathrm{\Omega }\subset {\mathbb{R}}^d$ (class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616303138&_mathId=si7.gif&_user=111111111&_pii=S0022039616303138&_rdoc=1&_issn=00220396&md5=e3906d187430441582915461a5328cab" title="Click to view the MathML source">d≥1class="mathContainer hidden">class="mathCode">$d\ge 1$) is a bounded smooth domain. We obtain new sufficient conditions for global existence and finite time blow-up of solutions, especially in the critical exponent cases: class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616303138&_mathId=si75.gif&_user=111111111&_pii=S0022039616303138&_rdoc=1&_issn=00220396&md5=a2b130994e7a9dd576012c56b4ab5d5f" title="Click to view the MathML source">p−1=rclass="mathContainer hidden">class="mathCode">$p-1=r$ and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616303138&_mathId=si9.gif&_user=111111111&_pii=S0022039616303138&_rdoc=1&_issn=00220396&md5=662242c9790f7f83b22b24bcb2055776" title="Click to view the MathML source">qr=(p−1)(s+1)class="mathContainer hidden">class="mathCode">$qr=(p-1)(s+1)$.