Local well-posedness and blow-up for an inhomogeneous nonlinear heat equation

In this paper, we consider the nonlinear heat equation with inhomogeneous nonlinearity (Formula presented.) where (Formula presented.) having either a polynomial growth or exponential growth, and (Formula presented.) is a function satisfying some assumptions to be stated later. We first prove the local well-posedness in suitable Lebesgue spaces when (Formula presented.) belongs to some Lebesgue space and (Formula presented.) has polynomial growth. We also obtain some blow-up results.