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Nonexistence Results for Semilinear Equations in Carnot Groups

Fausto Ferrari, Andrea Pinamonti (2013)

Analysis and Geometry in Metric Spaces

In this paper, following [3], we provide some nonexistence results for semilinear equations in the the class of Carnot groups of type ★.This class, see [20], contains, in particular, all groups of step 2; like the Heisenberg group, and also Carnot groups of arbitrarly large step. Moreover, we prove some nonexistence results for semilinear equations in the Engel group, which is the simplest Carnot group that is not of type ★.

Nonlinear evolution equations with exponential nonlinearities: conditional symmetries and exact solutions

Roman Cherniha, Oleksii Pliukhin (2011)

Banach Center Publications

New Q-conditional symmetries for a class of reaction-diffusion-convection equations with exponential diffusivities are derived. It is shown that the known results for reaction-diffusion equations with exponential diffusivities follow as particular cases from those obtained here but not vice versa. The symmetries obtained are applied to construct exact solutions of the relevant nonlinear equations. An application of exact solutions to solving a boundary-value problem with constant Dirichlet conditions...

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