Summary: This paper investigates some well-posedness issues of the fractional inhomogeneous Schrödinger equation \[ i\dot{u}-(-\Delta)^\gamma u=\pm |x|^\rho |u|^{p-1}u, \] where \(0 < \gamma < 1\) and \(\rho < 0\). Here, one considers the inter-critical regime \(0 < s_c := \frac{N}{2}-\frac{2\gamma + \rho}{p-1} < \gamma\), where \(s_c\) is the energy critical exponent, which is the only one real number satisfying \(\|\kappa^\frac{2\gamma+\rho}{p-1} u_0 (\kappa \cdot)\|_{\dot{H}^{s_c}} = \|u_0\|_{\dot{H}^{s_c}}\). In order to avoid a loss of regularity in Strichartz estimates, one assumes that the datum is spherically symmetric. First, using a sharp Gagliardo-Nirenberg-type estimate, one develops a local theory in the space \(\dot{H}^\gamma \cap \dot{H}^{s_c}\). Then, one investigates the \(L^{\frac{N(p-1)}{\rho+2\gamma}}\) concentration of finite-time blow-up solutions bounded in \(\dot{H}^{s_c}\). Finally, one proves the existence of non-global solutions with negative energy. Since one considers the homogeneous Sobolev space \(\dot{H}^{s_c}\), the main difficulty here is to avoid the mass conservation law.