### Mathematics in Engineering

2022, Issue 3: 1-24. doi: 10.3934/mine.2022024
Research article Special Issues

# A nonlinear diffusion equation with reaction localized in the half-line

• Received: 21 May 2021 Accepted: 19 July 2021 Published: 10 August 2021
• 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

### Related Papers:

• 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 $p = 1\neq m$. ###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142 1.0 2.2

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