| Abstract: |
| In this talk, we discuss the dichotomy between blow-up and global existence for solutions of the Cauchy problem for a heat equation with initial data in $H^1(\mathbb R^N)$. We consider non-homogeneous nonlinearities with polynomial growth (when $N\geq 3$) and exponential growth (when $N=2$) which are critical in the energy space $H^1(\mathbb R^N)$ according to the Sobolev and the Trudinger-Moser inequality respectively. By means of energy methods, we study the asymptotic behavior of solutions with low energies, and we show that the splitting between blow-up and global existence is determined by the sign of a suitable functional. |
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