The boundary degeneracy of a singular diffusion equation
Huashui Zhan, Qingmei Xie · Journal of Inequalities and Applications · 2014
We consider the following singular diffusion equation with boundary degeneracy: , , where is a bounded domain with appropriately smooth boundary, , , and . Though its diffusion coefficient vanishes on the boundary, it is still possible that the heat flux transfers across the boundary (Yin and Wang in Chin. Ann. Math., Ser. B 25:175-182, 2004), and it is not possible to define the homogeneous boundary value condition as usual. In the paper, under the assumption on the uniqueness of the weak solution, if the point x lies in the interior of the domain Ω, the paper obtains the result that the weak solution of the quoted equation has the same regular properties as those of the weak solution to the usual evolutionary p-Laplacian equation. However, if the point x lies on the boundary ∂ Ω, the situation may be different. The most significant feature of the paper is that the definition of the homogeneous boundary value condition of the above equation is given. Then, if , the bounded estimates of the weak solution are got by constructing the special barrier functions, and at last, how the diffusion coefficient affects the gradient of the solution near the boundary is discussed. MSC:35K55, 35K65, 35B40.