Semilinear Heat Inequalities with a Hardy-Type Potential in an Exterior Geodesic Domain on $\mathbb{S}^N$

Mohamed Jleli; Bessem Samet

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arXiv · 2605.09080

Semilinear Heat Inequalities with a Hardy-Type Potential in an Exterior Geodesic Domain on $\mathbb{S}^N$

Mohamed Jleli·Bessem Samet

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Abstract

We study an inhomogeneous semilinear heat inequality on the unit sphere $\mathbb S^N$, $N\ge3$, in an exterior geodesic domain associated with a fixed pole. The equation involves the singular Hardy-type potential $\lambda/\sin^2 r$, where $r=d(o,x)$, and the weighted nonlinearity $(\sin r)^\alpha |u|^p$. For $\alpha>-2$ and $0<\lambda\le \lambda^*=((N-2)/2)^2$, we prove the existence of a critical exponent $p_{\mathrm{crit}}=p_{\mathrm{crit}}(\alpha,N,\lambda)$ governing the existence and nonexistence of solutions. More precisely, we prove that no weak solution exists for any nontrivial nonnegative source in the range $p>p_{\mathrm{crit}}$, whereas classical solutions exist for some positive continuous sources in the range $1 1$. The analysis is based on the construction of radial Hardy barriers adapted to the antipodal singularity and on sharp integral estimates involving power and logarithmic cutoffs near $r=\pi$.

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Mohamed Jleli, Bessem Samet. 2026-05-09. Semilinear Heat Inequalities with a Hardy-Type Potential in an Exterior Geodesic Domain on $\mathbb{S}^N$. https://arxiv.org/abs/2605.09080

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