Blow-up solutions and global solutions to discrete p -laplacian parabolic equations

Abstract

We discuss the conditions under which blow-up occurs for the solutions of discrete p-Laplacian parabolic equations on networks S with boundary ∂S as follows: ut(x,t)=Δp,ωu(x,t)+|u(x,t)|q-1u(x,t), (x,t)εS×(0,+∞); u(x,t)=0, (x,t)ε∂S×(0,+∞); u(x,0)=u0≥0, xεS¯, where p>1, q>0, >0, and the initial data u0 is nontrivial on S. The main theorem states that the solution u to the above equation satisfies the following: (i) if 0 1, then the solution blows up in a finite time, provided u¯0>ω0/1/q-p+1, where ω0:=maxxεS∑yS¯ω(x,y) and u¯0:=maxxεS u0(x); (ii) if 0