The heat equation with a singular potential

Source abstract

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