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Blow-up for a non-local diffusion problem with Neumann boundary conditions and a reaction term. (English) Zbl 1169.35312

Summary: We study the blow-up problem for a non-local diffusion equation with a reaction term,
\[ u_t(x,t)= \int_\Omega J(x-y)(u(y,t)-u(x,t))\,dy+ u^p(x,t). \]
We prove that non-negative and non-trivial solutions blow up in finite time if and only if \(p>1\). Moreover, we find that the blow-up rate is the same as the one that holds for the ODE \(u_t=u^p\), that is, \(\lim_{t\nearrow T} (T-t)^{\frac{1}{p-1}} \|u(\cdot,t)\|_\infty= (\frac{1}{p-1})^{\frac{1}{p-1}}\). Next, we deal with the blow-up set. We prove single point blow-up for radially symmetric solutions with a single maximum at the origin, as well as the localization of the blow-up set near any prescribed point, for certain initial conditions in a general domain with \(p>2\). Finally, we show some numerical experiments which illustrate our results.

MSC:

35B40 Asymptotic behavior of solutions to PDEs
45G10 Other nonlinear integral equations
45K05 Integro-partial differential equations
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