A New Condition for the Concavity Method of Blow-up Solutions to Semilinear Heat Equations

Abstract

In this paper, we consider the semilinear heat equations under Dirichlet boundary condition \[ u_{t}\left(x,t\right)=Δu\left(x,t\right)+f(u(x,t)), & \left(x,t\right)\in Ω\times\left(0,+\infty\right), u\left(x,t\right)=0, & \left(x,t\right)\in\partial Ω\times\left[0,+\infty\right), u\left(x,0\right)=u_{0}\geq0, & x\in\overlineΩ, \] where $\Omega$ is a bounded domain of ${\mathbb{R}}^N$ $(N\ge 1)$ with smooth boundary $\partial \Omega$. The main contribution of our work is to introduce a new condition \[ (C) α\int_{0}^{u}f(s)ds \leq uf(u)+βu^{2}+γ,\,\,u>0 \] for some $\alpha ,\beta ,\gamma >0$ with $0<\beta \le \frac{\left(\alpha -2\right){\lambda }_0}{2}$, where ${\lambda }_0$ is the first eigenvalue of Laplacian $\Delta$, and we use the concavity method to obtain the blow-up solutions to the semilinear heat equations. In fact, it will be seen that the condition (C) improves the conditions known so far.

7 pages