Local and Global Analysis of Semilinear Heat Equations with Hardy Potential on Stratified Lie Groups

Abstract

On stratified Lie groups we study a semilinear heat equation with the Hardy potential, a power non-linearity and a forcing term which depends only upon the spacial variable. The analysis of an equivalent formulation to the problem and an application of a decade old result of Avelin et al. facilitates the management of the singularity in the Hardy potential, thereby yielding results pertaining to both local and global nonexistence. In addition, local existence is verified when the gradient term appearing in the Hardy potential is unimodular almost everywhere. The global existence is also proved under an additional assumption that the forcing term depends on the time variable as well. Through these results this paper sheds light on the possible pivotal exponents for the existence of both local and global solutions to the equation, offering a deeper understanding of the interplay between the model's parameters and the underlying stratified Lie group structure.