Classification of singular solutions of porous media equations with absorption

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Proceedings of the Royal Society of Edinburgh: Section A Mathematics, ISSN: 0308-2105, Vol: 135, Issue: 03, Page: 563-584

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Xinfu Chen; Yuanwei Qi; Mingxin Wang
Cambridge University Press (CUP)
Mathematics; HEAT-EQUATION; PARABOLIC EQUATIONS; Mathematics; Applied; Mathematics
article description
We consider, for m ∈ (0, 1) and q > 1, the porous media equation with absorption u= Δu- uin ℝ× (0, ∞). We are interested in those solutions, which we call singular solutions, that are non-negative, non-trivial, continuous in ℝ× (0, ∞)\{(0,0)}, and satisfy u(x,0) = 0 for all x ≠ 0. We prove the following results. When q ≥ m + 2/n, there does not exist any such singular solution. When q < m + 2/n, there exists, for every c > 0, a unique singular solution u = u, called the fundamental solution with initial mass c, which satisfies ∫u(.,t) → c as t ↘ 0. Also, there exists a unique singular solution u = u, called the very singular solution, which satisfies ∫u(.,t) → ∞ as ↘ 0. In addition, any singular solution is either uor ufor some finite positive c, u< uwhen c< c, and u↗ uas c↗ ∞. Furthermore, uis self-similar in the sense that u(x,t) = tw(|x|t) for α = 1/(q - 1), β= 1/2(q - m), and some smooth function w defined on [0, ∞), so that ∫u(.,t) is a finite positive constant independent of t > 0. © 2005 The Royal Society of Edinburgh.