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A critical non-homogeneous heat equation with weighted source
- Part of
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- Journal:
- European Journal of Applied Mathematics , First View
- Published online by Cambridge University Press:
- 06 March 2025, pp. 1-12
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- Article
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Some qualitative properties of radially symmetric solutions to the non-homogeneous heat equation with critical density and weighted source
are obtained, in the range of exponents
\begin{align*} |x|^{-2}\partial _tu=\Delta u+|x|^{\sigma }u^p, \quad (x,t)\in {\mathbb {R}}^N\times (0,T), \end{align*}
$p\gt 1$, 
$\sigma \ge -2$. More precisely, we establish conditions fulfilled by the initial data in order for the solutions to either blow-up in finite time or decay to zero as 
$t\to \infty$ and, in the latter case, we also deduce decay rates and large time behavior. In the limiting case 
$\sigma =-2$, we prove the existence of non-trivial, non-negative solutions, in stark contrast to the homogeneous case. A transformation to a generalized Fisher–KPP equation is derived and employed in order to deduce these properties.
