We study the behaviour of the solutions to the quasilinear heat equation with a reaction restricted to a half-line
$ u_t = (u^m)_{xx}+a(x) u^p, $
$ m, p > 0 $ and $ a(x) = 1 $ for $ x > 0 $, $ a(x) = 0 $ for $ x < 0 $. We first characterize the global existence exponent $ p_0 = 1 $ and the Fujita exponent $ p_c = m+2 $. Then we pass to study the grow-up rate in the case $ p\le1 $ and the blow-up rate for $ p > 1 $. In particular we show that the grow-up rate is different as for global reaction if $ p > m $ or $ p = 1\neq m $.
Citation: Raúl Ferreira, Arturo de Pablo. A nonlinear diffusion equation with reaction localized in the half-line[J]. Mathematics in Engineering, 2022, 4(3): 1-24. doi: 10.3934/mine.2022024
We study the behaviour of the solutions to the quasilinear heat equation with a reaction restricted to a half-line
$ u_t = (u^m)_{xx}+a(x) u^p, $
$ m, p > 0 $ and $ a(x) = 1 $ for $ x > 0 $, $ a(x) = 0 $ for $ x < 0 $. We first characterize the global existence exponent $ p_0 = 1 $ and the Fujita exponent $ p_c = m+2 $. Then we pass to study the grow-up rate in the case $ p\le1 $ and the blow-up rate for $ p > 1 $. In particular we show that the grow-up rate is different as for global reaction if $ p > m $ or $ p = 1\neq m $.
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Raúl Ferreira, Arturo de Pablo. A nonlinear diffusion equation with reaction localized in the half-line[J]. Mathematics in Engineering, 2022, 4(3): 1-24. doi: 10.3934/mine.2022024
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