Classification of singular solutions of porous media equations with absorption
 Citation data:

Royal Society of Edinburgh  Proceedings A, ISSN: 03082105, Vol: 135, Issue: 3, Page: 563584
 Publication Year:
 2005

 Scopus 6
 CrossRef 2
 Groups:
 Researchers:
 Xinfu Chen
 Repository URL:
 http://stars.library.ucf.edu/facultybib2000/5060; http://stars.library.ucf.edu/facultybib/7325
 DOI:
 10.1017/s0308210500004005
 Author(s):
 Publisher(s):
 Tags:
 Mathematics; HEATEQUATION; 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 nonnegative, nontrivial, 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 selfsimilar in the sense that u(x,t) = tw(xt) 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.