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| Formato: | Preprint |
| Publicado: |
2023
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| Acceso en línea: | https://arxiv.org/abs/2302.02063 |
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| _version_ | 1866911855534931968 |
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| author | Chen, Wenhui |
| author_facet | Chen, Wenhui |
| contents | We study semilinear third-order (in time) evolution equations with fractional Laplacian $(-Δ)^σ$ and power nonlinearity $|u|^p$, which was proposed by Bezerra-Carvalho-Santos [2] recently. In this manuscript, we obtain a new critical exponent $p=p_{\mathrm{crit}}(n,σ):=1+\frac{6σ}{\max\{3n-4σ,0\}}$ for $n\leqslant\frac{10}{3}σ$. Precisely, the global (in time) existence of small data Sobolev solutions is proved for the supercritical case $p>p_{\mathrm{crit}}(n,σ)$, and weak solutions blow up in finite time even for small data if $1<p\leqslant p_{\mathrm{crit}}(n,σ)$. Furthermore, to more accurately describe the blow-up time, we derive new and sharp upper bound as well as lower bound estimates for the lifespan in the subcritical case and the critical case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_02063 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Critical exponent and sharp lifespan estimates for semilinear third-order evolution equations Chen, Wenhui Analysis of PDEs We study semilinear third-order (in time) evolution equations with fractional Laplacian $(-Δ)^σ$ and power nonlinearity $|u|^p$, which was proposed by Bezerra-Carvalho-Santos [2] recently. In this manuscript, we obtain a new critical exponent $p=p_{\mathrm{crit}}(n,σ):=1+\frac{6σ}{\max\{3n-4σ,0\}}$ for $n\leqslant\frac{10}{3}σ$. Precisely, the global (in time) existence of small data Sobolev solutions is proved for the supercritical case $p>p_{\mathrm{crit}}(n,σ)$, and weak solutions blow up in finite time even for small data if $1<p\leqslant p_{\mathrm{crit}}(n,σ)$. Furthermore, to more accurately describe the blow-up time, we derive new and sharp upper bound as well as lower bound estimates for the lifespan in the subcritical case and the critical case. |
| title | Critical exponent and sharp lifespan estimates for semilinear third-order evolution equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2302.02063 |