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

Abstract

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

Keywords

• Discrete p-Laplacian
• Semilinear parabolic equation
• Blow-up