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It is known that the solution of equation $$ u_t - \triangle u = \kappa|u|^{\sigma}u, u(0) = u_0 $$ blow up in finite times if $\sigma > 0$. That is, the $L^{\infty}$ norm of solution $u$ will goes to $\infty$ as $t$ goes to $t_0$ for some finite time $t_0$. My question is what happens if we consider $L^2$ norm instead of $L^\infty$? Dose the $L^2$ norm of solution also blows up in finite time?

In my opinion, it may be exist a global solution in $L^2$. We consider $\sigma = \frac{4}{N}$ for brief. It is known that the above problem is local well posed in $L^2$ under this growing condition. On the other hand, multiplying the equation by $u$ and integrating on $R^N$ we find $$ \frac{1}{2}\frac{d}{dt} \|u\|^2 + \|\nabla u\|^2 = \int |u|^{2 + \frac{4}{N}} $$ where $\|\cdot\|$ denotes the norm of $L^2$. And since the Lieb-Thirring inequality $\int |u|^{2 + \frac{4}{N}} \leq C \|\nabla u\|^2$, which implies that $\|u\| \leq \|u_0\|$ for $t \leq T$ if we choose $\kappa$ small enough. That is , the $L^2$ norm of solution $u$ is finite for any given initial data $u_0$.

Is my conclusion correct?

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There are non-trivial global smooth solutions with small $L^2$-norm and small $L^\infty$-norm if $\sigma > \frac{2}{N}$, by results of H. Fujita (J. Fac. Science Univ. Tokyo, 1966). Fujita's original is paper is here:

All nontrivial non-negative solution blow up in finite time if $\sigma < \frac{2}{N}$. There has been a lot of work on this, see Google Scholar.

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This is not the case. The Lieb - THirring inequality holds only for $u \in H^1$ and $|u|_{L^2} = 1$.

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