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- W3001792232 abstract "In the present paper, we study the Cauchy problem for the wave equation with a time-dependent scale invariant damping, i.e.$frac{2}{1+t}partial_t v$ and a cubic convolution $(|x|^{-gamma}*v^2)v$ with $gammain (0,n)$, where $v=v(x,t)$ is an unknown function on $mathbb{R}^ntimes[0,T)$. Our aim of the present paper is to prove a small data blow-up result and show an upper estimate of lifespan of the problem for slowly decaying positive initial data $(v(x,0),partial_t v(x,0))$ such as $partial_t v(x,0)=O(|x|^{-(1+nu)})$ as $|x|rightarrowinfty$. Here $nu$ belongs to the scaling supercritical case $nu<frac{n-gamma}{2}$. Our main new contribution is to estimate the convolution term in high spatial dimensions, i.e. $nge 4$. This paper is the first blow-up result to treat wave equations with the cubic convolution in high spatial dimensions ($nge 4$)." @default.
- W3001792232 created "2020-01-30" @default.
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- W3001792232 date "2020-01-22" @default.
- W3001792232 modified "2023-10-16" @default.
- W3001792232 title "Small data blow-up for the wave equation with a time-dependent scale invariant damping and a cubic convolution for slowly decaying initial data" @default.
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- W3001792232 doi "https://doi.org/10.48550/arxiv.2001.07985" @default.
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