On the weighted Sobolev space and its application on the existence of weak solutions related to the p-Laplacian
Huashui Zhan · Research Square · 2023
Abstract The initial boundary value problem of an evolutionary equation with the form is considered, where a(x) ≥ 0 is a weight and may be degenerate on some points of Ω. Such a equation arises from the heat flow in a certain composite materia or reaction-diffusion problem in an environmental heterogeneity and barriers. In the paper, we assume that and invistigate the Sobolev inequality and the embedding inequality in the weighted Sobolev space are true. Then, by generalizing the the classical Gagliardo-Nirenberg inequality to the weighted Sobolev space, using the difference method as [10], the existence and uniqueness of weak solutions are obtained provided that p > 2. Mathematics Subject Classification: 35K65 35L85 35R92