EXISTENCE OF SOLUTIONS TO THE PARABOLIC EQUATION WITH A SINGULAR POTENTIAL OF THE SOBOLEV-HARDY TYPE

doi:10.1016/S0252-9602(12)60148-9

Acta mathematica scientia,Series B ›› 2012, Vol. 32 ›› Issue (5): 1901-1918.doi: 10.1016/S0252-9602(12)60148-9

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EXISTENCE OF SOLUTIONS TO THE PARABOLIC EQUATION WITH A SINGULAR POTENTIAL OF THE SOBOLEV-HARDY TYPE

HAN Jun-Qiang^*, WANG Yong-Da, NIU Peng-Cheng

Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an 710129, China

Received:2011-02-23 Revised:2010-05-07 Online:2012-09-20 Published:2012-09-20
Contact: HAN Jun-Qiang, pengchengniu@nwpu.edu.cn E-mail:southhan@163.com; wwwyda666@sina.com; pengchengniu@nwpu.edu.cn
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).

Abstract

Abstract:

We study the existence of solutions to the following parabolic equation
{ut − Δ_pu =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 −Δ_pu ≡ −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 C¹boundary ∂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.

Key words: nonlinear parabolic equations, existence, Sobolev-Hardy inequality, singular potential

CLC Number:

35K25

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.

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EXISTENCE OF SOLUTIONS TO THE PARABOLIC EQUATION WITH A SINGULAR POTENTIAL OF THE SOBOLEV-HARDY TYPE

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