A condition for blow-up solutions to discrete p-Laplacian parabolic equations under the mixed boundary conditions on networks

Abstract

AbstractIn this paper, we investigate the condition $$(C_{p})\quad \alpha \int _{0}^{u}f(s)\,ds \leq uf(u)+\beta u^{p}+\gamma ,\quad u>0 $$(Cp)α∫0uf(s)ds≤uf(u)+βup+γ,u>0 for some $\alpha >2$α>2, $\gamma >0$γ>0, and $0\leq \beta \leq \frac{ (\alpha -p ) \lambda _{p,0}}{p}$0≤β≤(α−p)λp,0p, where $p>1$p>1, and $\lambda _{p,0}$λp,0 is the first eigenvalue of the discrete p-Laplacian $\Delta _{p,\omega }$Δp,ω. Using this condition, we obtain blow-up solutions to discrete p-Laplacian parabolic equations $$ \textstyle\begin{cases} u_{t} (x,t )=\Delta _{p,\omega }u (x,t )+f(u(x,t)), & (x,t )\in S\times (0,+\infty ), \\ \mu (z)\frac{\partial u}{\partial _{p} n}(x,t)+\sigma (z) \vert u(x,t) \vert ^{p-2}u(x,t)=0, & (x,t )\in \partial S\times [0,+\infty ), \\ u (x,0 )=u_{0}\geq 0\quad (\mbox{nontrivial}), & x\in S, \end{cases} $${ut(x,t)=Δp,ωu(x,t)+f(u(x,t)),(x,t)∈S×(0,+∞),μ(z)∂u∂pn(x,t)+σ(z)|u(x,t)|p−2u(x,t)=0,(x,t)∈∂S×[0,+∞),u(x,0)=u0≥0(nontrivial),x∈S, on a discrete network S, where $\frac{\partial u}{\partial _{p}n}$∂u∂pn denotes the discrete p-normal derivative. Here μ and σ are nonnegative functions on the boundary ∂S of S with $\mu (z)+\sigma (z)>0$μ(z)+σ(z)>0, $z\in \partial S$z∈∂S. In fact, we will see that condition $(C_{p})$(Cp) improves the conditions known so far.