Large time behavior of general solutions to a class of quasilinear diffusion equations with a weighted source term \(\begin{aligned} \partial _tu=\Delta u^m+\varrho (x)u^p, \quad (x,t)\in \mathbb {R}^N\times (0,\infty ), \end{aligned}\) with \(m>1\) , \(1<p<m\) and suitable functions \(\varrho (x)\) , is established. More precisely, we consider functions \(\varrho \in C(\mathbb {R}^N)\) such that \(\begin{aligned} \lim \limits _{|x|\rightarrow \infty }(1+|x|)^{-\sigma }\varrho (x)=A\in (0,\infty ), \end{aligned}\) with \(\sigma \in (\max \{-N,-2\},0)\) such that \(L:=\sigma (m-1)+2(p-1)<0\) . We show that, for all these choices of \(\varrho \) , solutions with initial conditions \(u_0\in C(\mathbb {R}^N)\cap L^{\infty }(\mathbb {R}^N)\cap L^r(\mathbb {R}^N)\) for some \(r\in [1,\infty )\) are global in time and, if \(u_0\) is compactly supported, present the asymptotic behavior \(\begin{aligned} \lim \limits _{t\rightarrow \infty }t^{-\alpha }\Vert u(t)-V_*(t)\Vert _{\infty }=0, \end{aligned}\) where \(V_*\) is a suitably rescaled version of the unique compactly supported self-similar solution to the equation with the singular weight \(\varrho (x)=|x|^{\sigma }\) : \(\begin{aligned} U_*(x,t)=t^{\alpha }f_*(|x|t^{-\beta }), \qquad \alpha =-\frac{\sigma +2}{L}, \quad \beta =-\frac{m-p}{L}. \end{aligned}\) This behavior is an interesting example of asymptotic simplification for the equation with a regular weight \(\varrho (x)\) towards the singular one as \(t\rightarrow \infty \) .
