We study a wide class of space-time fractional equations with structural damping and nonlinear memory: \(\displaystyle \partial ^{2\alpha }_t u(t,x)+\mu (-\varDelta )^{\frac {\sigma }{2}}\partial ^\alpha _t u(t,x)+(-\varDelta )^\sigma u(t,x)=I^{1-\alpha }_{0^+}f(t,x), \) \(t\geq 0\) , \( x\in {\mathbf {R}}^n\) , where the parameters satisfying \( 1/2 <\alpha \leq 1\) , \(\mu >0\) and \(\sigma > 1\) . We take the so-called Caputo-Djrbashian derivative with respect to time t, \(\partial _t^{\kappa }\) of order \(\kappa \) with \(\kappa =2\alpha , \alpha \) , the fractional Laplacian in the space variable x, \((-\varDelta )^{\delta }\) with \(\delta =\frac {\sigma }{2},\sigma \) , and the Riemann-Liouville integral \(I^{1-\alpha }_{0^+}f(t,x)\) . We first obtain solution representations for Cauchy type problems expressed in terms of Mittag-Leffler’s functions \(E_{\alpha ,\beta }(z)\) ( \(z\in \mathbb {C}\) ) by using a Duhamel type formula with the aid of Laplace (in time) and Fourier (in space) transforms. Taking into account that the Fourier multipliers are radially symmetric functions, the convolution kernels are represented through the Hankel transform. Then, our purpose is to establish decay estimates of the derivatives of its solutions for initial data with \(L^p\) and Sobolev regularity. We highlight that the strategy to be employed relies on harmonic analysis tools. We also provide applications employing Strichartz’s estimates to prove existence results for mild solutions in particular cases, in which \(f(t,x)=| u(t,x) |^k\) with \(k >1\) or f belongs to a time weighted Banach space.