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Existence and uniqueness of a specific self-similar solution is establishedfor the following reaction-diffusion equation with Hardy singular potential $$\partial_tu=\Delta u^m+|x|^{-2}u^p, \qquad (x,t)\in \real^N\times(0,\infty), $$in the range of exponents $1\leq p<m$ and dimension $N\geq3$. The self-similarsolution is unbounded at $x=0$ and has a logarithmic vertical asymptote, but itremains bounded at any $x\neq0$ and $t\in(0,\infty)$ and it is a weak solutionin $L^1$ sense, which moreover satisfies $u(t)\in L^p(\real^N)$ for any $t>0$and $p\in[1,\infty)$. As an application of this self-similar solution, it isshown that there exists at least a weak solution to the Cauchy problemassociated to the previous equation for any bounded, nonnegative and compactlysupported initial condition $u_0$, contrasting with previous results inliterature for the critical limit $p=m$.

A special self-similar solution and existence of global solutions for a reaction-diffusion equation with Hardy potential