Blow-Up Solutions and Global Solutions to Discretep-Laplacian Parabolic Equations

摘要

We discuss the conditions under which blow-up occurs for the solutions of discretep-Laplacian parabolic equations on networksSwith boundary∂Sas follows:ut(x,t)=Δp,ωu(x,t)+λ|u(x,t)|q-1u(x,t),(x,t)∈S×(0,+∞);u(x,t)=0,(x,t)∈∂S×(0,+∞);u(x,0)=u0≥0,x∈S¯, wherep>1,q>0,λ>0, and the initial datau0is nontrivial onS. The main theorem states that the solutionuto the above equation satisfies the following: (i) if0<p-1<qandq>1, then the solution blows up in a finite time, providedu¯0>ω0/λ1/q-p+1, whereω0:=maxx∈S⁡∑y∈S¯‍ω(x,y)andu¯0:=maxx∈S u0(x); (ii) if0<q≤1, then the nonnegative solution is global; (iii) if1<q<p-1, then the solution is global. In order to prove the main theorem, we first derive the comparison principles for the solution of the equation above, which play an important role throughout this paper. Moreover, when the solution blows up, we give an estimate for the blow-up time and also provide the blow-up rate. Finally, we give some numerical illustrations which exploit the main results.