Does there exist a global solution in L^2 for reaction diffusion equation with focusing nonlinearity? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T00:37:20Z http://mathoverflow.net/feeds/question/59690 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/59690/does-there-exist-a-global-solution-in-l2-for-reaction-diffusion-equation-with-fo Does there exist a global solution in L^2 for reaction diffusion equation with focusing nonlinearity? Ming Wang 2011-03-27T02:00:34Z 2011-08-05T21:22:12Z <p>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?</p> <p>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 $u_0$.</p> <p>Is my conclusion correct?</p> http://mathoverflow.net/questions/59690/does-there-exist-a-global-solution-in-l2-for-reaction-diffusion-equation-with-fo/60292#60292 Answer by Ming Wang for Does there exist a global solution in L^2 for reaction diffusion equation with focusing nonlinearity? Ming Wang 2011-04-01T14:08:08Z 2011-04-01T14:08:08Z <p>This is not the case. The Lieb - THirring inequality holds only for $u \in H^1$ and $|u|_{L^2} = 1$.</p> http://mathoverflow.net/questions/59690/does-there-exist-a-global-solution-in-l2-for-reaction-diffusion-equation-with-fo/61826#61826 Answer by Hans Engler for Does there exist a global solution in L^2 for reaction diffusion equation with focusing nonlinearity? Hans Engler 2011-04-15T15:12:23Z 2011-04-15T15:12:23Z <p>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:</p> <p><a href="http://repository.dl.itc.u-tokyo.ac.jp/dspace/bitstream/2261/6061/1/jfs130201.pdf" rel="nofollow">http://repository.dl.itc.u-tokyo.ac.jp/dspace/bitstream/2261/6061/1/jfs130201.pdf</a></p> <p>All nontrivial non-negative solution blow up in finite time if $\sigma &lt; \frac{2}{N}$. There has been a lot of work on this, see Google Scholar.</p>