Nonstationary Discrete p-Laplacian

• Nonstationary discrete p-Laplacian is a framework modeling nonlinear diffusion-reaction phenomena on graphs via time-dependent (parabolic) equations.
• It establishes existence, uniqueness, and stabilization properties using subdifferential methods and discrete Sobolev spaces under varied boundary conditions.
• The analysis identifies threshold behaviors such as blow-up and extinction, with outcomes influenced by spectral properties and numerical convergence studies.

The nonstationary discrete $p$-Laplacian describes nonlinear diffusion-reaction phenomena on networks, governed by time-dependent (parabolic) equations featuring the discrete $p$-Laplacian operator. Unlike its stationary counterpart—corresponding to elliptic, time-independent problems—the nonstationary formulation models the evolution of functions over a graph subject to nonlinear diffusion and potentially superlinear or sublinear reactions. Key questions involve well-posedness, long-time behavior (including blow-up, extinction, or stabilization), and the influence of nonlinearity, boundary conditions, and network geometry.

1. Discrete $p$-Laplacian and Nonstationary Parabolic Framework

Let $G = (V, E, \omega)$ denote a (possibly infinite) weighted simple graph with symmetric weights $\omega(x, y) \geq 0$, and let $S \subset V$ be a finite or infinite node set, possibly with a boundary $\partial S$. The canonical form of the nonstationary discrete $p$-Laplace equation is

$$u_t(x, t) = \Delta_{p, \omega} u(x, t) + F(x, t, u), \qquad x \in S,$$

where, for $p > 1$, the discrete $p$-Laplacian acts as

$$\Delta_{p, \omega} u(x) = \sum_{y \sim x} \omega(x, y)\,|u(y) - u(x)|^{p-2} (u(y) - u(x)),$$

and $F$ encodes sources, reactions, or inhomogeneities.

Boundary conditions may be of Neumann, Dirichlet, or mixed (Robin) type, often modeled by combinations of discrete $p$-normal derivatives and local terms at $\partial S$. The initial data $u(x,0)$ is typically required to be nonnegative.

2. Existence, Uniqueness, and Comparison Principles

Well-posedness for the initial-boundary-value nonstationary discrete $p$-Laplacian equation on finite or infinite graphs is established via subdifferential theory and convex analysis in discrete Sobolev-like spaces. For finite graphs, local or global existence and uniqueness of strong solutions is obtained for $p > 1$ under mild data assumptions, often using a Schauder fixed-point framework and comparison principles (Hwang, 2019, Mugnolo, 2012). In the infinite graph case, appropriate choices of weighted $\ell^q$ and energy spaces support existence and uniqueness, and semigroup generators are characterized as subdifferentials of convex functionals (Hua et al., 2014, Mugnolo, 2012).

Order-preserving and $\ell^\infty$-contractive properties of the associated nonlinear semigroups ensure that positivity of the initial data propagates forward in time, and solutions preserve network symmetries induced by automorphisms (Mugnolo, 2012).

3. Blow-up, Extinction, and Global Existence: Parameter Regimes

The full taxonomy of solution regimes is articulated for the discrete $p$-Laplacian with $q$-reaction and mixed boundary conditions on finite graphs. The archetypal initial-boundary-value problem is: $$\begin{cases} u_t(x, t) = \Delta_{p, \omega} u(x, t) + \lambda |u(x, t)|^{q-1} u(x, t), & (x, t) \in S \times (0, \infty), \ \mu(z) \frac{\partial u}{\partial_p n}(z, t) + \sigma(z) |u(z, t)|^{p-2} u(z, t) = 0, & (z, t) \in \partial S \times [0, \infty), \ u(x, 0) = u_0(x) \geq 0, & x \in \overline{S}, \end{cases}$$ with $p > 1$, $q > 0$, $\lambda > 0$, $\mu, \sigma \geq 0$, and $\mu(z) + \sigma(z) > 0$ (Hwang, 2019).

The regimes are controlled by relations between $p$, $q$, $\lambda$, the initial datum, and the structure of the boundary:

• Neumann ($\sigma \equiv 0$): If $q > 1$, all nontrivial solutions blow up in finite time. For $q \leq 1$, solutions exist globally (with polynomial/exponential growth).
• Mixed/Robin ($\sigma \not\equiv 0$):
  • Super-supercritical ($q > 1$, $q > p-1 > 0$): Sufficiently large initial data induces finite-time blow-up; explicit thresholds and blow-up rates are given via the spectral properties ($d_\omega(x)$, $\lambda_{p,0}$).
  • Subcritical ($0 < q < p-1$): All solutions remain uniformly bounded and exist globally.
  • Critical case ($q = p-1 > 0$): The threshold $\lambda = \lambda_{p,0}$ separates finite-time blow-up, global boundedness, or vanishing, depending on subcases determined by $p$ and $q$.

Generic global existence and extinction results for infinite graphs depend on network isoperimetry and summability properties of the initial data; finite extinction is provable for $p < d+1$ under ($d$)-isoperimetric control (Hua et al., 2014). Conservation of mass applies for $p \geq 2$ in Neumann settings and positive initial data.

4. Large-Time Asymptotics, Stabilization, and Universal Bounds

On infinite graphs with inhomogeneous densities $\rho(x)$ and suitable Sobolev/isoperimetric properties, large-time behavior is controlled by sharp energy-decay and discrete embedding inequalities (Tedeev, 24 Dec 2025, Hua et al., 2014). For $p > 2$, stabilization rates for the solution supremum norm are derived explicitly, with rates depending on the decay of $\rho$ and volume growth. If $\rho(x)$ exhibits non-power decay, time-algebraic decay of $\|u(t)\|_\infty$ is obtained, often modified by logarithmic corrections:

• On $\mathbb{Z}^N$ with $\rho(x) \simeq |x|^{-\alpha}$, $0 < \alpha < p < N$:

$$\|u(t)\|_\infty \leq C\, M(0)^{(p-\alpha)/H} t^{-(N-\alpha)/H},\quad H = (N-\alpha)(p-2) + p - \alpha.$$

• For $\rho$ decaying fast enough ($\alpha > p$), a universal decay bound $\|u(t)\|_\infty \leq C t^{-1/(p-2)}$ is established, independent of initial data.

Finite graphs with finite measure and Poincaré inequality exhibit algebraic decay to the network mean for $p > 2$ (Hua et al., 2014).

5. Spectral and Variational Characterization; Threshold Phenomena

The classification of blow-up and extinction thresholds relies on the first eigenvalue $\lambda_{p,0}$ of $-\Delta_{p, \omega}$ under the prescribed boundary conditions. Variationally,

$$\lambda_{p,0} = \inf_{\sum_{x \in S} \mu(x) |\phi(x)|^p = 1} \left\{ \frac{1}{2}\sum_{x, y \in S} |\phi(x) - \phi(y)|^p\,\omega(x, y) + \sum_{z \in \partial S} \sigma(z) |\phi(z)|^p \right\}.$$

Criticality with respect to $\lambda_{p,0}$ appears, for example, in the mixed boundary case with $q = p-1$, separating blow-up, global existence, and extinction (Hwang, 2019).

The $C_p$-condition in blow-up analysis unifies and extends prior continuous/probabilistic criteria, taking into account both the nonlinearity in $u$ and the spectral gap $\lambda_{p,0}$ (Chung et al., 2019). The explicit involvement of the network’s spectrum refines thresholds for finite-time blow-up and improves known criteria.

6. Numerical and Approximation Theory

Nonstationary discrete $p$-Laplacian problems are amenable to numerical simulation and error analysis. Well-posed time-discretization (explicit/implicit Euler) is established for general nonlocal kernels and arbitrary $p$, with quantitative convergence rates as the graph (e.g., via a graphon limit) approximates a continuum domain (Yosra et al., 2016). The limit $p \to \infty$ yields evolution within Cheeger-type sublevel sets. Contraction properties are preserved under graph discretizations, supporting stability and accuracy in practical computations.

Numerical experiments on small graphs confirm the theoretical blow-up criteria, threshold behavior, and asymptotic rates predicted by the spectral and analytic theory (Hwang, 2019).

7. Connections, Generalizations, and Open Directions

The discrete nonstationary $p$-Laplacian framework parallels and informs its continuous counterpart, offering precise theorems for finite and infinite graphs that can serve both as prototypes and test cases for nonlinear PDE behavior. Generalizations include variable-exponent $p$-Laplacians, nonlocal kernels, signless and normalized Laplacians, and discrete Schrödinger operators (Mugnolo, 2012). The extension to graphs with measure-theoretic boundary at infinity (Martin or Royden boundaries) provides further analytic subtlety, especially in infinite or random graphs.

A plausible implication is that the detailed regimes uncovered in the discrete, finite-network case—particularly the sharp role of the spectrum and initial profile—may guide analogous classification and threshold theory for reaction-diffusion and nonlinear diffusion equations in Euclidean and manifold settings.