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Analyticity and regularity for a class of second order evolution equations

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  • The regularity conservation as well as the smoothing effect are studied for the equation $ u''+ Au+ cA^\alpha u' = 0$, where $A$ is a positive selfadjoint operator on a real Hilbert space $H$ and $\alpha\in (0, 1]; \,\, c >0$. When $\alpha\ge {1\over 2}$ the equation generates an analytic semigroup on $D(A^{1/2})\times H $ , and if $\alpha\in (0, {1\over 2})$ a weaker optimal smoothing property is established. Some conservation properties in other norms are also established, as a typical example, the strongly dissipative wave equation $u_{tt} - \Delta u -c\Delta u_t = 0$ with Dirichlet boundary conditions in a bounded domain is given, for which the space $C_0(\Omega)\times C_0(\Omega)$ is conserved for $t>0$, which presents a sharp contrast with the conservative case $u_{tt} - \Delta u = 0$ for which $C_0(\Omega)$-regularity can be lost even starting from an initial state $(u_0, 0)$ with $u_0\in C_0(\Omega)\cap C^1(\overline {\Omega})$.
    Mathematics Subject Classification: 35B65, 35D99, 35L05, 35L90.


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