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 p>1$p>1$, s>−1$s>-1$, 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">$q,r,d_1,d_2,a_1,a_2$ are positive constants, 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">${\delta }_1,{\delta }_2,u_0,v_0$ are nonnegative smooth functions, Ω⊂R^d$\mathrm{\Omega }\subset {\mathbb{R}}^d$ (d≥1$d\ge 1$) 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: p−1=r$p-1=r$ and qr=(p−1)(s+1)$qr=(p-1)(s+1)$.