Fujita Exponent for the Fractional Sub-Laplace Semi-Linear Heat Equation With Forcing Term on the Heisenberg Group

Oza P. Suragan D.
2025 John Wiley and Sons Ltd

Mathematical Methods in the Applied Sciences
2025

We study the following semi-linear heat equation in the Heisenberg group (Formula presented.) : (Formula presented.) where (Formula presented.) denotes the fractional sub-Laplacian of order (Formula presented.) on (Formula presented.). We establish that the Fujita exponent, the critical threshold that delimits different dynamical regimes of this equation, is (Formula presented.) where (Formula presented.) is the homogeneous dimension of (Formula presented.). We prove the existence of global-in-time solutions for the supercritical case (Formula presented.) and the nonexistence of global-in-time solutions for the subcritical case (Formula presented.) For the critical case (Formula presented.) we provide a class of functions for which (Formula presented.) blows up in finite time. These results extend the classical Fujita phenomenon to a sub-Riemannian setting with the nonlocal effects of the fractional sub-Laplacian. Our proof methods intertwine analytic techniques with the geometric structure of the Heisenberg group.

finite time blow-up , fractional Laplacian , Fujita exponent , global existence , Heisenberg group , integro-PDEs

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Department of Mathematics, Nazarbayev University, Astana, Kazakhstan

Department of Mathematics

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