Heat equations driven by mixed local-nonlocal operators with exponential nonlinearity
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| author | Chaurasia, Dharmendra Kumar Fino, Ahmad Z. Kumar, Vishvesh |
| author_facet | Chaurasia, Dharmendra Kumar Fino, Ahmad Z. Kumar, Vishvesh |
| contents | We investigate the Cauchy problem for a heat equation driven by the mixed local-nonlocal operator $\mathcal{L}:=-Δ+(-Δ)^s$, $s\in(0,1)$, with exponential nonlinearity \[ \partial_tu(x,t)+\mathcal{L}u(x,t)=f(u(x,t)), \qquad (x,t)\in \mathbb{R}^{d}\times(0,\infty), \] where $f:\mathbb{R}\to\mathbb{R}$ exhibits exponential growth at infinity and satisfies $f(0)=0$. We establish local well-posedness in a suitable Orlicz space in the case where $f(u)\sim e^{|u|^p}$ as $|u|\to\infty$, with $p>1$. We further prove the existence of global solutions for small initial data under the assumption that $f$ satisfies the growth condition $|f(u)|\sim |u|^m$ near the origin. Moreover, we derive large-time decay estimates in Lebesgue spaces, showing that the behavior of the nonlinearity near the origin determines the decay rate of solutions and highlights a unique asymptotic transition that bridges local and non-local diffusion theories. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_03657 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Heat equations driven by mixed local-nonlocal operators with exponential nonlinearity Chaurasia, Dharmendra Kumar Fino, Ahmad Z. Kumar, Vishvesh Analysis of PDEs 35K58, 35B33, 35A01, 35B44 We investigate the Cauchy problem for a heat equation driven by the mixed local-nonlocal operator $\mathcal{L}:=-Δ+(-Δ)^s$, $s\in(0,1)$, with exponential nonlinearity \[ \partial_tu(x,t)+\mathcal{L}u(x,t)=f(u(x,t)), \qquad (x,t)\in \mathbb{R}^{d}\times(0,\infty), \] where $f:\mathbb{R}\to\mathbb{R}$ exhibits exponential growth at infinity and satisfies $f(0)=0$. We establish local well-posedness in a suitable Orlicz space in the case where $f(u)\sim e^{|u|^p}$ as $|u|\to\infty$, with $p>1$. We further prove the existence of global solutions for small initial data under the assumption that $f$ satisfies the growth condition $|f(u)|\sim |u|^m$ near the origin. Moreover, we derive large-time decay estimates in Lebesgue spaces, showing that the behavior of the nonlinearity near the origin determines the decay rate of solutions and highlights a unique asymptotic transition that bridges local and non-local diffusion theories. |
| title | Heat equations driven by mixed local-nonlocal operators with exponential nonlinearity |
| topic | Analysis of PDEs 35K58, 35B33, 35A01, 35B44 |
| url | https://arxiv.org/abs/2605.03657 |