On the paper "On the Cauchy Problem for Reaction-Diffusion Equations" Wang studies the Hardy-Hénon equation $$ \begin{cases} u_t - \Delta u = |\cdot|^{l}u^{p}& \mbox{ in } \mathbb{R}^n \times (0,T), \\ {u}(0) = {u}_{0}& \mbox{ in } \mathbb{R}^n \\ \end{cases}. $$ On page 554 he says that "when $l<0$ it is generally impossible to obtain a classical solution for the equation". I was curious if there really isn't any classical solution, or if there aren't mathematical methods yet to determine whether or not there is a classical solution. In the case $l<0$ it deals only with mild solutions. Has there been any progress in the literature in this direction?
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2$\begingroup$ Well, there's an obvious obstruction: if $u$ is a classical (i.e. $C^2$) solution, and $u > 0$ (this is preserved if the initial condition is positive by the maximum principle, for example), and $l < 0$, then you have a contradiction. The left hand side is bounded, but the right hand side blows up at $x = 0$ (I assume the $\cdot$ is $x$ here?). So if you are going to look for a solution, you shouldn't expect a $C^2$ one. $\endgroup$– user378654Commented May 19, 2023 at 1:21
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$\begingroup$ Got it, I imagined he can't use standard parabolic regularity results, so he came to this conclusion. $\endgroup$– IlovemathCommented May 19, 2023 at 14:52
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