The heat equation with a singular potential

Abstract:

Of concern is the singular problem $\partial u/\partial t = \Delta u + (c/|x{|^2}) u + f(t,x), u(x,0) = u_{0}(x)$, and its generalizations. Here $c \geqslant 0,x \in {{\mathbf {R}}^N},t > 0$, and $f$ and ${u_0}$ are nonnegative and not both identically zero. There is a dimension dependent constant ${C_{\ast } }(N)$ such that the problem has no solution for $c > {C_{\ast } }(N)$. For $c \leqslant {C_{\ast } }(N)$ necessary and sufficient conditions are found for $f$ and ${u_0}$ so that a nonnegative solution exists.