Lifespan estimate for the semilinear regular Euler-Poisson-Darboux-Tricomi equation
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866916597151563776 |
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| author | Li, Yuequn Guo, Fei |
| author_facet | Li, Yuequn Guo, Fei |
| contents | In this paper, we begin by establishing local well-posedness for the semilinear regular Euler-Poisson-Darboux-Tricomi equation. Subsequently, we derive a lifespan estimate with the Strauss index given by $p=p_{S}(n+\fracμ{m+1}, m)$ for any $δ>0$, where $δ$ is a parameter to describe the interplay between damping and mass. This is achieved through the construction of a new test function derived from the Gaussian hypergeometric function and a second-order ordinary differential inequality, as proven by Zhou \cite{Zhou2014}. Additionally, we extend our analysis to prove a blow-up result with the index $p=\max\{p_{S}(n+\fracμ{m+1}, m), p_{F}((m+1)n+\frac{μ-1-\sqrtδ}{2})\}$ by applying Kato$^{\prime}$s Lemma ( i.e., Lemma \ref{katolemma} ), specifically in the case of $δ=1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_02084 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Lifespan estimate for the semilinear regular Euler-Poisson-Darboux-Tricomi equation Li, Yuequn Guo, Fei Analysis of PDEs 35B44, 35G25 I.7.1 In this paper, we begin by establishing local well-posedness for the semilinear regular Euler-Poisson-Darboux-Tricomi equation. Subsequently, we derive a lifespan estimate with the Strauss index given by $p=p_{S}(n+\fracμ{m+1}, m)$ for any $δ>0$, where $δ$ is a parameter to describe the interplay between damping and mass. This is achieved through the construction of a new test function derived from the Gaussian hypergeometric function and a second-order ordinary differential inequality, as proven by Zhou \cite{Zhou2014}. Additionally, we extend our analysis to prove a blow-up result with the index $p=\max\{p_{S}(n+\fracμ{m+1}, m), p_{F}((m+1)n+\frac{μ-1-\sqrtδ}{2})\}$ by applying Kato$^{\prime}$s Lemma ( i.e., Lemma \ref{katolemma} ), specifically in the case of $δ=1$. |
| title | Lifespan estimate for the semilinear regular Euler-Poisson-Darboux-Tricomi equation |
| topic | Analysis of PDEs 35B44, 35G25 I.7.1 |
| url | https://arxiv.org/abs/2502.02084 |