Provided the non-negative function
g Î Lloc1(W)g \in L_{\rm loc}^1(\Omega)
allows for a generalized Hardy-Sobolev inequality, existence and uniqueness of global weak solutions of the possibly degenerate parabolic PDE
g(x)[(u)\dot]=Dug(x)\dot{u}=\Delta u
, subject to homogeneous Dirichlet boundary conditions, is proved. The maximum/minimum principle holds. The associated entropy decays exponentially as
t
with a rate not exceeding 2/C, where
C is the constant arising in the generalized Hardy-Sobolev inequality.
2000 Mathematics Subject Classification: 46E35, 35K65, 35B05, 35B40, 35B50
Key words: Hardy-Sobolev inequality, degenerate parabolic PDE, existence and uniqueness of global solutions, maximum principle, minimum principle, exponential decay of entropy
A.U. acknowledges support from the DFG Forschungszentrum 
Mathematics for Key Technologies
, project D10 (Berlin) and from the EU Research Network HYKE. M.R. acknowledges the hospitality of the mathematical department, Universität Kaiserslautern, where this work was carried out.