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| Main Author: | |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2302.02063 |
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Table of 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.