Inhomogeneous parabolic equations on unbounded metric measure spaces
  We study inhomogeneous semilinear parabolic equations with source term f independent of time u_{t}={\Delta}u+u^{p}+f(x) on a metric measure space, subject to the conditions that f(x)\geq 0 and u(0,x)=\phi(x)\geq 0. By establishing Harnack-type inequalities in time t and some powerful estimates, we give sufficient conditions for non-existence, local existence, and global existence of weak solutions. This paper generalizes previous results on Euclidean spaces to general metric measure spaces. 
Kenneth J. Falconer, Jiaxin Hu, and Yuhua Sun
math-ph math.MP