A New Condition for the Concavity Method of Blow-up Solutions to p-Laplacian Parabolic Equations

In this paper, we consider an initial-boundary value problem of the p-Laplacian parabolic equations \begin{equation} \begin{cases} u_{t}\left(x,t\right)=\mbox{div}(|\nabla u\left(x,t\right)|^{p-2}\nabla u(x,t))+f(u(x,t)), & \left(x,t\right)\in \Omega\times\left(0,+\infty\right), \newline u\left(x,t\right)=0, & \left(x,t\right)\in\partial \Omega\times\left[0,+\infty\right), \newline u\left(x,0\right)=u_{0}\geq0, & x\in\overline{\Omega}, \end{cases} \end{equation} where $p\ge 2$ and $\mathrm{\Omega }$ is a bounded domain of ${\mathbb{R}}^N$ $(N\ge 1)$ with smooth boundary $\mathrm{\partial }\mathrm{\Omega }$. The main contribution of this work is to introduce a new condition $(C_{p)$$\hspace{1cm} \alpha \int_{0}^{u}f(s)ds \leq uf(u)+\beta u^{p}+\gamma,\,\,u>0$} for some $\alpha ,\beta ,\gamma >0$ with $0<\beta \le \frac{(\alpha -p){\lambda }_{1,p}}{p}$, where $\lambda_{1, p}$isthefirsteigenvalueofp-Laplacian$\Delta_{p}$, and we use the concavity method to obtain the blow-up solutions to the above equations. In fact, it will be seen that the condition $(C_p)$ improves the conditions ever known so far.