Inhomogeneous parabolic equations on unbounded metric measure spaces

In recent years, the study of partial differential equations on self-similar fractals has attracted increasing interest (see, for example, [7–9, 13, 14]). We investigate a class of nonlinear diffusions with source terms on general metric measure spaces. Diffusion is of fundamental importance in many areas of physics, chemistry and biology. Applications of diffusion include sintering, i.e. making solid materials from powder (powder metallurgy, production of ceramics); catalyst design in the chemical industry; diffusion of steel (e.g. with carbon or nitrogen) to modify its properties; doping during production of semiconductors. Let (M,d, μ) be a metric measure space, that is, (M,d) is a locally compact separable metric space and μ is a Radon measure on M with full support. We consider the following nonlinear diffusion equation with a source term f on (M,d, μ):