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A NOTE ON THE INFLUENCE OF DIFFERENT ADDITIONAL REGULARITY ON THE CRITICAL EXPONENT
1 : Laboratory of Analysis and Control of PDEs
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Djillali Liabes University, P.O. Box 89, 22000 Sidi Bel Abbes, Algeria -
Algeria
In this paper, we consider the Cauchy problem for the semi-linear damped $\sigma$-evolution equations, where the initial data are supposed to belong to the energy space with different additional regularity, which means that,
$$(u_{0}, u_{1}) \in\Big(H^{\sigma}(\mathbb{R}^{n})\cap L^{m_{1}}(\mathbb{R}^{n}) \Big)\times\left( L^{2}(\mathbb{R}^{n})\cap L^{m_{2}}(\mathbb{R}^{n})\right), \ \ m_{1}, m_{2}\in [1,2), \ \sigma\geq 1.$$
Our main goal is to study the influence of $m_{1}$ and $m_{2}$ on the critical exponent by proving the global (in time)
existence of small data energy solutions where their decay estimates coincide with those to the corresponding linear equation.
