Analyticity and regularity for a class of second order evolution equations

Abstract

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 domainis 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 $u_0\in C_0(\Omega)\cap C^1(\overline {\Omega})$.