Finite time blow-up for a heat equation in RN

We consider the semilinear heat equation $u_t-\Delta u={\left|u\right|}^{p-1}u+\lambda u,\text{on}{R}^n$where $p>1$, and $\lambda \in R$ is a parameter. When $\lambda =0$, the equation reduces to the classical heat equation. We reveal that the parameter $\lambda$ in the linear term plays an important role in the blow-up conditions. Although the solution may blow up in finite time due to the cumulative effect of the nonlinearities, interestingly, we find that for $\lambda >\frac{n}{2}$, all non-negative solutions blow up in finite time, which shows that the Fujita exponent is equal to $+\infty$. Our result extends the Theorem 17.1 in Quittner and Souplet (2007). In addition, for $\lambda <0$, we provide a new sufficient condition for the finite time blow-up solution.