Non-existence of nonnegative separate variable solutions to a porous medium equation with spatially dependent nonlinear source

TL;DR

This paper proves the non-existence of certain nonnegative solutions to a porous medium equation with spatially dependent nonlinear source for large enough source strength, providing optimal bounds in higher dimensions and improved bounds in lower dimensions.

Contribution

It establishes new non-existence results for solutions to a porous medium equation with spatially dependent sources, including optimal bounds in high dimensions and improved bounds in lower dimensions.

Findings

Non-existence of solutions for large source strength in high dimensions.

Optimal lower bound on source parameter in dimensions N≥4.

Improved bounds on source parameter in dimensions N=1,2,3.

Abstract

The non-existence of nonnegative compactly supported classical solutions to $$- \Delta V(x) - |x|^\sigma V(x) + \frac{V^{1/m}(x)}{m-1} = 0, \qquad x\in\mathbb{R}^N,$$ with $m>1$, $\sigma>0$, and $N\ge 1$, is proven for $\sigma$ sufficiently large. More precisely, in dimension $N\geq4$, the optimal lower bound on $\sigma$ for non-existence is identified, namely $$\sigma\geq\sigma_c := \frac{2(m-1)(N-1)}{3m+1},$$ while, in dimensions $N\in\{1,2,3\}$, the lower bound derived on $\sigma$ improves previous ones already established in the literature. A by-product of this result is the non-existence of nonnegative compactly supported separate variable solutions to a porous equation medium equation with spatially dependent superlinear source.