Title

Singular Solutions Of Parabolic P-Laplacian With Absorption

Keywords

Absorption; Fast diffusion; Fundamental solution; P-Laplacian; Very singular solution

Abstract

We consider, for p ∈ (1, 2) and q > 1, the p-Laplacian evolution equation with absorption ut = div(|∇u|p-2∇u)- uq in ℝn × (0,∞). We are interested in those solutions, which we call singular solutions, that are non-negative, non-trivial, continuous in ℝn × [0,∞) \ {(0, 0)}, and satisfy u(x, 0) = 0 for all x ≠ 0. We prove the following: (i) When q ≥ p-1 + p/n, there does not exist any such singular solution. (ii) When q < p-1+p/n, there exists, for every c > 0, a unique singular solution u = uc that satisfies ∫ℝn u(·, t) → c as t ↘ 0. Also, uc ↗ u∞ as c ↗ ∞, where u∞ is a singular solution that satisfies ∫ℝn u∞(·, t)→∞ as t ↘ 0. Furthermore, any singular solution is either u∞ or u c for some finite positive c. © 2007 American Mathematical Society.

Publication Date

11-1-2007

Publication Title

Transactions of the American Mathematical Society

Volume

359

Issue

11

Number of Pages

5653-5668

Document Type

Article

Personal Identifier

scopus

DOI Link

https://doi.org/10.1090/S0002-9947-07-04336-X

Socpus ID

77950178814 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/77950178814

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