Extinction and asymptotic behavior of solutions for the ω-heat equation on graphs with source and interior absorption

Blow-up problems for nonlinear parabolic equations on locally finite graphs

“…As is known to us, many structures in our real life can be represented by a connected graph whose vertices represent nodes and whose edges represent their links, such as the internet, brain, organizations and so on. In recent years, the investigations of discrete weighted Laplacians and various equations on graphs have attracted attention from many authors (see [1,4,5,9,10,11,12,16,18,20] and references therein). There have been some works on dealing with blow-up phenomenon of the equations on graphs, for example, Xin et al [20] investigated the blow-up properties of the Dirichlet boundary value problem for u t = ∆u + u p (p > 0) on a finite graph.…”

Convergence of ground state solutions for nonlinear Schrödinger equations on graphs

“…There are works related to some important geometric inequalities on graphs [3,20]. Many aspects about heat equation such as existence and non-existence of global solutions [7,24,25], uniqueness and blow-up properties [21,30], estimates for heat kernel [20,23,29] have also been considered. For the elliptic case, Grigoryan et al [16][17][18] established existence results on graphs for some nonlinear elliptic equations based on the variational framework and this inspires us to deal with the Schödinger type equations on graphs.…”

Existence and convergence of solutions for nonlinear biharmonic equations on graphs

“…To do analysis works, it is necessary to study partial differential equations on graphs and this subject has attracted much attention recently. For example, several fundamental aspects of heat equations on graphs, such as heat kernel [14,29], existence and uniqueness [16,20,22] are investigated by different authors.…”