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$ L^p $-$ L^q $ estimates for the damped wave equation and the critical exponent for the nonlinear problem with slowly decaying data

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  • 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 $ 1\le q \le p < \infty\ (p\neq 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^s\cap H_r^{\beta}) \times (H^{s-1} \cap L^r) $ with $ r \in (1,2] $, $ s\ge 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 by an ODE argument.

    Mathematics Subject Classification: Primary: 35L71; Secondary: 35A01, 35B40, 35B44.


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