Extinction and positivity of the solutions for a \(p\)-Laplacian equation with absorption on graphs. (English) Zbl 1228.65154

Summary: We deal with the extinction of the solutions of the initial-boundary value problem of the discrete \(p\)-Laplacian equation with absorption \(u_t = \Delta_{p,\omega} u - u^q\) with \(p > 1\), \(q > 0\), which is said to be the discrete \(p\)-Laplacian equation on weighted graphs. For \(0 < q < 1\), we show that the nontrivial solution becomes extinction in finite time, while it remains strictly positive for \(p \geq 2\), \(q\geq 1\) and \(q\geq p-1\). Finally, a numerical experiment on a simple graph with standard weight is given.


65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35K55 Nonlinear parabolic equations
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