Acta mathematica scientia, Series B >
EXISTENCE OF SOLUTIONS TO THE PARABOLIC EQUATION WITH A SINGULAR POTENTIAL OF THE SOBOLEV-HARDY TYPE
Received date: 2011-02-23
Revised date: 2010-05-07
Online published: 2012-09-20
Supported by
Research was supported by NPU Foundation for Fun-damental Research (NPU-FFR-JC201124), NSF of China (10871157, 11001221, 11002110), and Specialized Research Fund for the Doctoral Program in Higher Education (200806990032).
We study the existence of solutions to the following parabolic equation
{ut − Δpu =λ/|x|s |u|q−2 u, (x, t) ∈ Ω × (0,∞),
u(x, 0) = f(x), x ∈Ω,
u(x, t) = 0, (x, t) ∈ ∂Ω × (0,∞), (P)
where −Δpu ≡ −div(|∇u|p−2 ∇u), 1 < p < N, 0 < s ≤ p, p ≤ q ≤ p*(s) = N−s/N−p p, Ω is a bounded domain in RN such that 0 ∈ Ω with a C1 boundary ∂Ω, f ≥ 0 satisfying some convenient regularity assumptions. The analysis reveals that the existence of solutions for (P) depends on p, q, s in general, and on the relation between and the best constant in the Sobolev-Hardy inequality.
HAN Jun-Qiang , WANG Yong-Da , NIU Peng-Cheng . EXISTENCE OF SOLUTIONS TO THE PARABOLIC EQUATION WITH A SINGULAR POTENTIAL OF THE SOBOLEV-HARDY TYPE[J]. Acta mathematica scientia, Series B, 2012 , 32(5) : 1901 -1918 . DOI: 10.1016/S0252-9602(12)60148-9
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