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Multiple blow-up for a porous medium equation with reaction

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The present paper is concerned with the Cauchy problem

$$\left\{\begin{array}{ll}\partial_t u = \Delta u^m + u^p & \quad {\rm in}\; \mathbb R^N \times (0,\infty),\\ u(x,0) = u_0(x) \geq 0 & \quad {\rm in}\; \mathbb R^N, \end{array}\right.$$

with p, m > 1. A solution u with bounded initial data is said to blow up at a finite time T if \({{\lim {\rm sup}_{t \nearrow T}||u(t)||_{L^\infty(\mathbb{R}^N)} =\infty}}\). For N ≥ 3 we obtain, in a certain range of values of p, weak solutions which blow up at several times and become bounded in intervals between these blow-up times. We also prove a result of a more technical nature: proper solutions are weak solutions up to the complete blow-up time.

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Mizoguchi, N., Quirós, F. & Vázquez, J.L. Multiple blow-up for a porous medium equation with reaction. Math. Ann. 350, 801–827 (2011). https://doi.org/10.1007/s00208-010-0584-5

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