Cheung, Ka-Luen; Zhang, Zhenyu Nonexistence of global solutions for a family of nonlocal or higher order parabolic problems. (English) Zbl 1265.35174 Differ. Integral Equ. 25, No. 7-8, 787-800 (2012). The authors prove nonexistence of a global solution to the equation \(u_t=\Delta u+f(u)--\kern -9.5pt\int _{\Omega }f(u)\), \(t>0\), \(x\in \Omega \subset \mathbb {R}^n\), satisfying initial data \(u_0\) and Neumann boundary data. It is supposed: if \(u_0\) has the zero mean, \(\frac {1}{2}\int _{\Omega }| \nabla u_0| ^2-\int _{\Omega }F(u_0)\) is negative and \(\int _{\Omega }| \nabla u_0| ^2-\int _{\Omega }u_0f(u_0)\) is less than a given constant. Furthermore, some results on global nonexistence of the solution to the the initial boundary value problem to the equation \(u_t+(-1)^{m-1}\Delta ^m(\Delta u+f(u))=0\) (\(m\geq 1\)) are presented. Reviewer: Marie Kopáčková (Praha) Cited in 3 Documents MSC: 35K58 Semilinear parabolic equations 35B44 Blow-up in context of PDEs 35K20 Initial-boundary value problems for second-order parabolic equations 35K35 Initial-boundary value problems for higher-order parabolic equations Keywords:parabolic equation; nonexistence; global solution; potential well × Cite Format Result Cite Review PDF