On the nonexistence of global solutions for nonlocal parabolic equations with forcing terms

The purpose of this work is to analyze the well-posedness and blow-up behavior of solutions to the nonlocal semilinear parabolic equation with a forcing term:$ \partial_t u - \Delta u = \|u(t)\|_{q}^\alpha |u|^p + t^{\varrho} \mathbf{w}(x) \quad \text{in} \quad \mathbb{R}^N \times (0, \infty), $where $ N \geq 1 $, $ p, q \geq 1 $, $ \alpha \geq 0 $, $ \varrho > -1 $, and $ \mathbf{w}(x) $ is a suitably given continuous function.The novelty of this work, compared to previous studies, lies in considering a nonlocal nonlinearity $ \|u(t)\|_{q}^\alpha |u|^p $ and a forcing term $ t^{\varrho} \mathbf{w}(x) $ that depend on both time and space variables. This combination introduces new challenges in understanding the interplay between the nonlocal structure of the equation and the spatio-temporal forcing term.Under appropriate assumptions, we establish the global existence of solutions for small initial data in Lebesgue spaces when the exponent $ p $ exceeds a critical value. In contrast, we show that the global existence cannot hold for $ p $ below this critical value, provided the additional condition $ \int_{\mathbb{R}^N} \mathbf{w}(x) \, dx > 0 $ is satisfied. The main challenge in this analysis lies in managing the complex interaction between the nonlocal nonlinearity and the forcing term, which significantly influences the behavior of solutions.