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Nonexistence of global solutions for a family of nonlocal or higher order parabolic problems. (English) Zbl 1265.35174

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.

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
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