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

Abstract

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 Ω\times\left(0,+\infty\right), \newline u\left(x,t\right)=0, & \left(x,t\right)\in\partial Ω\times\left[0,+\infty\right), \newline u\left(x,0\right)=u_{0}\geq0, & x\in\overlineΩ, \end{cases} \end{equation} where $p\ge 2$ and $\Omega$ is a bounded domain of ${\mathbb{R}}^N$ $(N\ge 1)$ with smooth boundary $\partial \Omega$. The main contribution of this work is to introduce a new condition \[ \mbox{$(C_p)$$\alpha {\int }_0^{u}f(s)ds\le uf(u)+\beta u^p+\gamma ,u>0$} \] for some $\alpha ,\beta ,\gamma >0$ with $0<\beta \le \frac{\left(\alpha -p\right){\lambda }_{1,p}}{p}$, where ${\lambda }_{1,p}$ is the first eigenvalue of p-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.

15 pages. arXiv admin note: text overlap with arXiv:1706.03494

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