Generalized Hardy-Sobolev inequalities and exponential decay of the entropy of \(g(x)\dot u=\Delta u\) (Q1884780)

Let \(\Omega \subset \mathbb{R}^d \) be a smooth bounded domain and \(g \in L^1_{\text{loc}}(\Omega)\) be a non-negative function which satisfies the following generalized Hardy-Sobolev inequality: \[ \int_{\Omega} v^2 g \,dx \leq C \int_{\Omega} | \nabla v| ^2 \,dx,\quad \forall v \in H^1_o(\Omega), tag 1 \] where \(C \in \mathbb{R}^+\) is a constant which only depends on \(\Omega.\) The authors consider the (possibly degenerated) parabolic equation \(g \frac{\partial u}{\partial t} = \Delta u \) in \((0,t) \times \Omega,\) associated with the homogeneous Dirichlet boundary conditions on the boundary of \(\Omega,\) and the initial condition \(u| _{t=0} =u_0 \in L^2 (g \,dx)\). They prove the existence and the uniqueness of a weak solution of this problem and the validity of the maximum-minimum principle. They study the behavior of the associated entropy \(E(t),\) as \(t \to \infty,\) and show the exponential decay law \(E(t)\leq E(0) e^{-\frac{2t}{C}},\) where \(C\) is the constant in (1). An analogous exponential decay of the square gradient norm is also obtained. The main tools in the proofs, are the properties of the Lebesgue-measure, of the Lebesgue-Sobolev spaces and of the L. Schwartz 's theory of distributions