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• W3100346616 abstract "We study the Cauchy problem of the damped wave equation begin{align*} partial_{t}^2 u - Delta u + partial_t u = 0 end{align*} and give sharp $L^p$-$L^q$ estimates of the solution for $1le q le p < infty (pneq 1)$ with derivative loss. This is an improvement of the so-called Matsumura estimates. Moreover, as its application, we consider the nonlinear problem with initial data in $(H^scap H_r^{beta}) times (H^{s-1} cap L^r)$ with $r in (1,2]$, $sge 0$, and $beta = (n-1)|frac{1}{2}-frac{1}{r}|$, and prove the local and global existence of solutions. In particular, we prove the existence of the global solution with small initial data for the critical nonlinearity with the power $1+frac{2r}{n}$, while it is known that the critical power $1+frac{2}{n}$ belongs to the blow-up region when $r=1$. We also discuss the asymptotic behavior of the global solution in supercritical cases. Moreover, we present blow-up results in subcritical cases. We give estimates of lifespan and blow-up rates by an ODE argument." @default.
• W3100346616 created "2020-11-23" @default.
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• W3100346616 date "2019-01-01" @default.
• W3100346616 modified "2023-10-10" @default.
• W3100346616 title "$ L^p $-$ L^q $ estimates for the damped wave equation and the critical exponent for the nonlinear problem with slowly decaying data" @default.
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• W3100346616 doi "https://doi.org/10.3934/cpaa.2019090" @default.
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