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

2022/02/28 by Razvan Gabriel Iagar, Iagar, Razvan Gabriel, Philippe Laurençot +1
Mathematics · Computer Science · #Nonlinear Partial Differential Equations #Advanced Mathematical Physics Problems #Advanced Mathematical Modeling in Engineering

paper · pdf · doi:10.48550/arxiv.2202.13738

Abstract

The non-existence of nonnegative compactly supported classical solutions to - ΔV(x) - |x|^σV(x) + \fracV^1/m(x)m-1 = 0, x∈ℝ^N, with m>1, σ>0, and N≥ 1, is proven for σ sufficiently large. More precisely, in dimension N≥4, the optimal lower bound on σ for non-existence is identified, namely σ≥σ_c := (2(m-1)(N-1))/(3m+1), while, in dimensions N∈\1,2,3\, the lower bound derived on σ 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.

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