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

详细信息    查看全文

• 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 039616303138&_mathId=si2.gif&_user=111111111&_pii=S0022039616303138&_rdoc=1&_issn=00220396&md5=eccdbcedc1600e12c601cad7e75699b7" title="Click to view the MathML source">p>1$p>1$, 039616303138&_mathId=si3.gif&_user=111111111&_pii=S0022039616303138&_rdoc=1&_issn=00220396&md5=15b3183b91a40cf31a46eb53f25215ec" title="Click to view the MathML source">s>−1$s>-1$, 039616303138&_mathId=si4.gif&_user=111111111&_pii=S0022039616303138&_rdoc=1&_issn=00220396&md5=68954519f06eaa3bdcc7191080f4f240">039616303138-si4.gif">$q,r,d_1,d_2,a_1,a_2$ are positive constants, 039616303138&_mathId=si5.gif&_user=111111111&_pii=S0022039616303138&_rdoc=1&_issn=00220396&md5=786f5675b19b27d9c988e2f3e41f32f0">039616303138-si5.gif">${\delta }_1,{\delta }_2,u_0,v_0$ are nonnegative smooth functions, 039616303138&_mathId=si6.gif&_user=111111111&_pii=S0022039616303138&_rdoc=1&_issn=00220396&md5=89ce441b683e055992000ceb4291830c" title="Click to view the MathML source">Ω⊂R^d$\mathrm{\Omega }\subset {\mathbb{R}}^d$ (039616303138&_mathId=si7.gif&_user=111111111&_pii=S0022039616303138&_rdoc=1&_issn=00220396&md5=e3906d187430441582915461a5328cab" title="Click to view the MathML source">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: 039616303138&_mathId=si75.gif&_user=111111111&_pii=S0022039616303138&_rdoc=1&_issn=00220396&md5=a2b130994e7a9dd576012c56b4ab5d5f" title="Click to view the MathML source">p−1=r$p-1=r$ and 039616303138&_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)$qr=(p-1)(s+1)$.