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We investigate the Cauchy problem for the focusing inhomogeneous nonlinearSchr\"odinger equation $i \partial_t u + \Delta u = - |x|^b |u|^{p-1} u$ in theradial Sobolev space $H^1_{\text{rad}}(\mathbb{R}^N)$, where $b>0$ and $p>1$.We show the global existence and energy scattering in the inter-criticalregime, i.e., $p>\frac{N+4+2b}{N}$ and $p<\frac{N+2+2b}{N-2}$ if $N\geq 3$. Wealso obtain blowing-up solutions for the mass-critical and mass-supercriticalnonlinearities. The main difficulty, coming from the spatial growingnonlinearity, is overcome by refined Gagliardo-Nirenberg type inequalities. Ourproofs are based on improved Gagliardo-Nirenberg inequalities, theMorawetz-Sobolev approach of Dodson and Murphy, radial Sobolev embeddings, andlocalized virial estimates.

Long time dynamics and blow-up for the focusing inhomogeneous nonlinear Schr\"odinger equation with spatial growing nonlinearity