| Abstract: |
| In the first part of the talk, on unimodular Lie groups, we study the global well-posedness of the heat equation with a time-dependent nonlinearity of the form $S_p(t)f(u)$. We obtain distinct nonexistence criteria according to the volume growth of the group (compact, polynomial, or exponential). In the specific case of the Heisenberg group $\mathbb{H}^n$, we also provide sufficient conditions; for the case of the first Heisenberg group $\mathbb{H}^1$ these coincide with the necessary ones.
In the second part, we consider the same equation after dropping the group structure: the underlying space is $\mathbb{R}^n$ endowed with vector fields defining the operator and satisfying, among other assumptions, H\ormander's condition. In both settings, when $f(u)=u^p$, we compute the so-called critical Fujita exponent. |
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