  We consider the MGT equation with memory $$\partial_{ttt} u + \alpha
\partial_{tt} u - \beta \Delta \partial_{t} u - \gamma\Delta u +
\int_{0}^{t}g(s) \Delta u(t-s) ds = 0.$$ We prove an existence and uniqueness
result removing the convexity assumption on the convolution kernel $g$, usually
adopted in the literature. In the subcritical case $\alpha\beta>\gamma$, we
establish the exponential decay of the energy, without leaning on the classical
differential inequality involving $g$ and its derivative $g'$, namely,
$$g'+\delta g\leq 0,\quad\delta>0,$$ but only asking that $g$ vanishes
exponentially fast.
