We study the existence and long-time behavior of solutions to the following semilinear degenerate parabolic equation on $mathbb{R}^N$: $$frac{{partial}u}{{partial}t}-div({sigma}(x){ abla}u+{lambda}u+f(u)=g(x)$$, under a new condition concerning a variable non-negative diffusivity ${sigma}({cdot})$. Some essential difficulty caused by the lack of compactness of Sobolev embeddings is overcome here by exploiting the tail-estimates method.