We consider the semilinear heat equation ut−Δu=|u|p−1u+λu,onRnwhere p>1, and λ∈R is a parameter. When λ=0, the equation reduces to the classical heat equation. We reveal that the parameter λ 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 λ>n2, all non-negative solutions blow up in finite time, which shows that the Fujita exponent is equal to +∞. Our result extends the Theorem 17.1 in Quittner and Souplet (2007). In addition, for λ<0, we provide a new sufficient condition for the finite time blow-up solution.

中文翻译：

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[公式省略] 中热方程的有限时间放大


我们考虑半线性热方程 ut−Δu=|u|p−1u+λu，onRn其中 p>1，λ∈R 是一个参数。当 λ=0 时，方程简化为经典的热方程。我们揭示了线性项中的参数 λ 在爆炸条件下起着重要作用。虽然由于非线性的累积效应，解可能会在有限的时间内爆炸，但有趣的是，我们发现对于 λ>n2，所有非负解都在有限的时间内爆炸，这表明藤田指数等于 +∞。我们的结果扩展了 Quittner 和 Souplet （2007） 中的定理 17.1。此外，对于 λ<0，我们为有限时间放大解提供了新的充分条件。