In this paper, we consider the nonlinear heat equation with inhomogeneous nonlinearity u t − Δu = a(x) (u) where ∶ R → R having either a polynomial growth or exponential growth, and a ∶ R N → R is a function satisfying some assumptions to be stated later. We first prove the local well-posedness in suitable Lebesgue spaces when a belongs to some Lebesgue space and has polynomial growth. We also obtain some blow-up results. KEYWORDSblow-up, differential inequalities, existence, nonlinear heat equation, uniqueness MSC CLASSIFICATION 35K05; 35A01; 35B44Math Meth Appl Sci. 2020;43:5264-5272. wileyonlinelibrary.com/journal/mma

The purpose of this work is to analyze the blow-up of solutions of a nonlinear parabolic equation with a forcing term depending on both time and space variables \[u_t-\Delta u=|x|^{\alpha} |u|^{p}+{\mathtt a}(t)\,{\mathbf w}(x)\quad\text{for }(t,x)\in(0,\infty)\times \mathbb{R}^{N},\] where \(\alpha\in\mathbb{R}\), \(p\gt 1\), and \({\mathtt a}(t)\) as well as \({\mathbf w}(x)\) are suitable given functions. We generalize and somehow improve earlier existing works by considering a wide class of forcing terms that includes the most common investigated example \(t^\sigma\,{\mathbf w}(x)\) as a particular case. Using the test function method and some differential inequalities, we obtain sufficient criteria for the nonexistence of global weak solutions. This criterion mainly depends on the value of the limit \(\lim_{t\to\infty} \frac{1}{t}\,\int_0^t\,{\mathtt a}(s)\,ds\). The main novelty lies in our treatment of the nonstandard condition on the forcing term.