On global solutions of heat equations with time-dependent nonlinearities on unimodular Lie groups
arxiv(2024)
摘要
In this work, we study the global well-posedeness of the heat equation with
variable time-dependent nonlinearity of the form φ(t)f(u) on unimodular
Lie groups when the differential operator arises as the sum of squares of
Hörmander vector fields. For general unimodular Lie groups, we derive the
necessary conditions for the nonexistence of global positive solutions. This
gives different conditions in the cases of compact, polynomial, and exponential
volume growth groups. In the case of the Heisenberg groups ℍ^n, we
also derive sufficient conditions, which coincide with the necessary ones in
the case of ℍ^1 (and this is also true for ℝ^n). In
particular, in the case of the Heisenberg group ℍ^1 we obtain the
necessary and sufficient conditions under which the aforesaid initial value
problem with variable nonlinearity has a global positive solution.
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