Blow‐Up Solutions and Global Solutions to Discrete p‐Laplacian Parabolic Equations

Soon-Yeong Chung and Min-Jun Choi

Abstract and Applied Analysis, 2014, vol. 2014, issue 1

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) + 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¯, where p > 1, q > 0, λ > 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/λ1/q-p+1, where ω0:=maxx∈S⁡ ∑y∈S¯ ω(x,y) and u¯0:=maxx∈S u0(x); (ii) if 0

Date: 2014
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https://doi.org/10.1155/2014/351675

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Persistent link: https://EconPapers.repec.org/RePEc:wly:jnlaaa:v:2014:y:2014:i:1:n:351675

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