*Mathematical Methods in the Applied Sciences*A structurally damped σ-evolution equation with nonlinear memory

In this paper we investigate the global existence of small data
solutions for the following structurally damped σ-evolution model with
nonlinear memory term %
\[u_{tt}+(-\Delta)^\sigma
u+\mu(-\Delta)^{\frac{\sigma}{2}}u_t=\int_0^t
(1+\tau)^{-\gamma}|u_t(\tau,\cdot)|^p\,d\tau,\]
% with σ>0. In particular, for
$\gamma\in
((n-\sigma)/n,1)$ we find the sharp critical exponent,
under the assumption of small data in~$L^1$.
Dropping the~$L^1$ smallness assumption of initial
data, we show how the critical exponent is consequently modified for the
problem. In particular, we obtain a new interplay between the fractional
order of integration~$1-\gamma$ in the
nonlinear memory term, and the assumption that initial data are small
in~$L^m$, for
some~$m>1$.

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