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Blow-up phenomena for a nonlocal semilinear parabolic equation with positive initial energy. (English) Zbl 1443.35086

Summary: This paper is concerned with the blow-up of solutions to the following semilinear parabolic equation: \[ u_t=\Delta u+|u|^{p-1}u-\frac{1}{|\Omega|}\int_\Omega|u|^{p-1}udx,\quad x\in\Omega,\quad t>0, \] under homogeneous Neumann boundary condition in a bounded domain \(\Omega\subset\mathbb R^n\), \(n\geq 1\), with smooth boundary.
For all \(p>1\), we prove that the classical solutions to the above equation blow up in finite time when the initial energy is positive and initial data is suitably large. This result improves a recent result by W. Gao and Y. Han [Appl. Math. Lett. 24, No. 5, 784–788 (2011; Zbl 1213.35131)] which asserts the blow-up of classical solutions for \(n\geq 3\) provided that \(1<p\leq\frac{n+2}{n-2}\).

MSC:

35K91 Semilinear parabolic equations with Laplacian, bi-Laplacian or poly-Laplacian
35B44 Blow-up in context of PDEs
35K20 Initial-boundary value problems for second-order parabolic equations
35R09 Integro-partial differential equations

Citations:

Zbl 1213.35131
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References:

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