Critical exponent and sharp lifespan estimates for semilinear third-order evolution equations
Authors
Wenhui Chen
Categories
Abstract
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.
Critical exponent and sharp lifespan estimates for semilinear third-order evolution equations
Categories
Abstract
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.
Authors
Wenhui Chen
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