Uniqueness in Degenerate Parabolic Equations

• Uniqueness classes for degenerate parabolic equations are defined by weighted function spaces and precise integral conditions that control solutions near singularities.
• Methodologies such as weighted energy estimates, monotonicity principles, and viscosity solution techniques ensure the uniqueness of solutions under degeneracy.
• These frameworks have practical implications in physical, biological, and financial models by providing rigorous criteria for well-posedness despite boundary or interior degeneracies.

Degenerate parabolic equations are a class of evolution PDEs in which the diffusivity or other coefficients may vanish or become singular on part of the boundary or within the domain, leading to a loss of classical parabolicity or ellipticity in critical regions. This phenomenon complicates well-posedness theory: standard uniqueness or maximum principles often fail, and solutions require precise regularity, growth, or integral control to guarantee uniqueness. Uniqueness classes are rigorously specified function spaces or integral conditions providing sharp criteria that separate admissible (unique) solutions from pathological/non-unique ones for each degenerate operator and boundary regime.

1. Core Operator Classes and Degeneracy Structures

Degenerate parabolic problems arise in both divergence and non-divergence formulations, in linear and nonlinear settings, and include a variety of boundary or interior degeneracy scenarios:

• Linear divergence-form with vanishing diffusion: $\partial_t u = \operatorname{div}(a(x,t)\nabla u)$, with $a(x,t)\sim d(x)^\gamma$ near $\partial\Omega$ and $d(x)=\operatorname{dist}(x,\partial\Omega)$; no boundary conditions imposed (Nobili et al., 2020).
• Degenerate $p$-Laplacian: $$\partial_t u - v^{-1}\operatorname{div}(|\sqrt{Q}\nabla u|^{p-2}Q\nabla u)=0$$, where $v\geq 0$ (weight), $Q$ degenerate or singular (Cruz-Uribe et al., 22 May 2024).
• Nonlinear Fokker-Planck and McKean–Vlasov flows: $$\rho_t - \Delta\beta(\rho) + \nabla \cdot (D b(\rho)\rho)=0$$ with degenerate nonlinearities $\beta$ (Barbu et al., 2022).
• Degenerate/singular by boundary densities: $x_d^\alpha$-weighted parabolic problems in half-space, with $\alpha>-1$ (Dong et al., 2020), or $x_d^2$-weighted coefficients (Dong et al., 2 Sep 2025).
• Boundary-degenerate fully nonlinear viscosity problems: $\partial_t u + H(x,u,Du,D^2u) = 0$, $H$ degenerates near boundary, generating implicit (state-constraint) boundary conditions (Liu et al., 17 Jun 2024).
• Mixed/abstract and nonlinear degenerate parabolic equations: with variable coefficients and nonlocal or nonstandard boundary conditions, e.g., on weighted anisotropic Sobolev or Lions spaces (Shakhmurov et al., 2017, Punzo et al., 2014, Rassias et al., 2013).

The degeneracy may be encoded as power-law vanishing of the diffusion near boundaries, vanishing of the matrix weight, or through more general singular and weighted structures.

2. Characterization of Uniqueness Classes

The uniqueness of solutions for degenerate parabolic operators is not guaranteed by classical assumptions; instead, it is achieved via function space restrictions and/or boundary growth/integral conditions:

2.1 Weighted $L^1$ Uniqueness Classes

For linear equations of the form $\partial_t u = \operatorname{div}(a(x,t)\nabla u)$ with $a(x,t) \sim d(x)^\gamma$, uniqueness in the class of possibly unbounded solutions is enforced by one of two integral conditions, depending on $\gamma$ (Nobili et al., 2020):

Regime	Degeneracy	Integral Condition (Moment Type)	Equivalent Weighted Space
$\gamma>2$	Very strong	$$\int_{0}^T \!\int_{\Omega\setminus \Omega^\epsilon} |u| \leq C e^{\theta\epsilon^{2-\gamma}}$$	$$L^1_\phi(Q_T),~\phi(x) = e^{-\theta d(x)^{2-\gamma}}$$
$1\leq\gamma\leq 2$	Moderate	$$\int_{0}^T\!\int_{\Omega^{\epsilon/2}\setminus \Omega^{2\epsilon/3}} |u|\,d(x)^{\gamma-2} \leq C\epsilon^{\mu}$$, $\mu>-2\gamma+4$	$L^1_\phi(Q_T),~\phi(x) = d(x)^{\gamma-2-\mu}$

A solution with zero initial data and satisfying the corresponding integral restriction must vanish identically. For inhomogeneous problems, uniqueness within the same integral class is enforced.

2.2 Weighted Energy and Sobolev Spaces

Weighted Sobolev spaces are pivotal for uniqueness in degenerate operators with singular/weighted coefficients:

• For $p$-Laplacian degeneracies, solutions are unique in classes $u\in W^{1,2}_\mathrm{loc}((0,T); L^2_v)$, with $v$ and $Q$ only assuming minimal local integrability and two-sided degeneracy controls; no boundary conditions are assumed (Cruz-Uribe et al., 22 May 2024).
• Weak solutions to divergence/non-divergence equations are uniquely characterized in $W^{1,p}_*(\Omega_T, \beta)$ or $W^{1,2}_p(Q_T, x_d^\alpha)$ for weights $\beta$ or $x_d^\alpha$ in Muckenhoupt or similar classes (Fang et al., 22 Oct 2025, Dong et al., 2020).

2.3 Viscosity and Barrier-type Characterizations

In fully nonlinear (or infinity-Laplacian) problems, uniqueness is guaranteed for continuous viscosity solutions subject to appropriate continuity and growth at the parabolic boundary. No explicit boundary condition may be needed—degeneracy enforces an implicit (state-constraint) condition (Liu et al., 17 Jun 2024, Portilheiro et al., 2010). Classical comparison principles and barrier function constructions realize this mechanism in both nonlinear/porous medium and infinity-parabolic equations (Punzo et al., 2014, Portilheiro et al., 2010).

2.4 Spectral and Integral Representation: Sharpness and Loss of Uniqueness

Several frameworks provide explicit conditions for (non-)uniqueness:

• In the Heston degenerate PDE, uniqueness in the Tikhonov–Täcklind class together with vanishing faster than $1/v$ as $v\to0$ is sharp: examples show loss of uniqueness otherwise (Boyko, 14 Nov 2025).
• For interior degeneracy points (e.g., $k_a(x) = |x-a|$), uniqueness depends on spectral properties of the initial data (nonzero first Bessel coefficient), with explicit characterization of non-uniqueness in terms of resonance relations (Cannarsa et al., 2023).
• In equations with strong degeneracy and nonlocal boundary conditions, uniqueness (or non-uniqueness) is reduced to inequalities for parameters ($|a|<1$ and $\operatorname{Re} A\geq 0$ in the “a–b–c method”) (Rassias et al., 2013).

3. Essential Methodologies in Uniqueness Proofs

Methodologies for establishing uniqueness classes are diverse and tailored to the degenerate structure:

• Weighted Energy Methods and Test Functions: Multiplier-based integration with precise spatial cut-off and barrier functions handles boundary layers, boundary vanishing, and weighted energy dissipation (Nobili et al., 2020, Dong et al., 2020).
• Monotonicity Methods and Comparison Principles: Monotonicity of the degenerate/nonlinear operator flux in weighted Sobolev spaces and comparison principles in the viscosity framework underpin uniqueness, even for weak/mild/distributional solutions (Cruz-Uribe et al., 22 May 2024, Portilheiro et al., 2010, Barbu et al., 2022).
• Spectral Analysis and Series Expansion: For degenerate coefficients with internal singularities or interfaces, Bessel function expansions and associated monotonicity of spectral coefficients allow explicit identification of uniqueness or resonance-induced nonuniqueness (Cannarsa et al., 2023).
• State-constraint and Implicit Boundary Conditions: For fully nonlinear equations with boundary degeneracy, the analysis of test function touch points yields implicit (dynamic) boundary conditions in the viscosity sense, enforcing uniqueness without explicit boundary data (Liu et al., 17 Jun 2024).
• Fixed Point and Maximal Regularity Schemes: In abstract and high-regularity settings (nonlocal boundary, Banach-space–valued problems), $R$-sectoriality and analytic semigroup theory are used to construct unique solutions in weighted anisotropic Sobolev–Lions spaces (Shakhmurov et al., 2017).

4. Interplay with Boundary and Growth Conditions

The boundary behavior of solutions is central to uniqueness in degenerate settings:

• For purely interior degenerate diffusion, integral "smallness at the boundary" substitutes for Dirichlet or Neumann boundary data, specifying decay of solution mass near the degeneracy locus (Nobili et al., 2020).
• For Fokker-Planck flows and degenerate $p$-Laplacian, physically relevant boundary growth or decay is automatic from integrability and weight conditions; for degenerate/singular equations with Robin or Dirichlet boundary, the uniqueness class necessitates precise trace or energy control (Fang et al., 22 Oct 2025, Thiam et al., 2023, Punzo et al., 2014).
• In the context of option pricing (Heston model), sublinear blow-up at the ellipticity-degenerate boundary is the critical threshold; loss of uniqueness occurs even if the solution is globally well-controlled but fails this near-boundary restriction (Boyko, 14 Nov 2025).
• Fully nonlinear problems: degeneracy at the boundary acts as an implicit dynamic boundary, preventing the existence of nontrivial extensions and thus ensuring uniqueness up to the boundary (Liu et al., 17 Jun 2024).

5. Sharpness, Optimality, and Breakdown of Uniqueness

The uniqueness conditions identified for degenerate parabolic equations are in general sharp:

• For linear equations with power-law degeneracy at the boundary, the moment/integral decay is both necessary and sufficient; construction of counterexamples (either explicit or via separation of variables/Bessel or polynomial representations) demonstrates failure of uniqueness if the boundary layer is not sufficiently controlled (Nobili et al., 2020, Rassias et al., 2013).
• The criticality of weighted Sobolev exponents or parameter ranges is established by duality, optimal Hardy inequalities, or direct construction of nontrivial solutions at extremal weights (Dong et al., 2020, Dong et al., 2 Sep 2025).
• For viscosity solutions, loss of uniqueness corresponds to failures of the comparison principle at the boundary, e.g., in the case $p<2$ for heat-type boundary degeneracies (Liu et al., 17 Jun 2024, Portilheiro et al., 2010).
• In inverse problems, uniqueness for internal degeneracy identification hinges on nonvanishing spectral data; sign-changing or vanishing coefficients in the initial data lead to ambiguity in the reconstructed parameter (Cannarsa et al., 2023).

6. Connections to Broader Uniqueness Theories and Applications

The uniqueness classes for degenerate parabolic equations link analytical theory to practical modeling and applications:

• Physical and Financial Models: Degenerate equations naturally model diffusion in materials with vanishing conductivity, population genetics (Kimura operator), porous medium flow, and option pricing models with stochastic volatility (Heston model) (Boyko, 14 Nov 2025).
• Dependence on Data and Parameters: Uniqueness classes often reveal the precise regularity, decay, or monotonicity required of initial data, weights, or functional parameters (such as matrix weights, Robin boundary data, or interaction coefficients) (Barbu et al., 2022, Cannarsa et al., 2023).
• Generalizations and Open Problems: Current sharp characterizations prompt generalizations to fractional/parabolic operators, Laplacians on fractals, time-dependent degeneracies, and nonsmooth domains (Liu et al., 17 Jun 2024).
• Methodological Innovations: The field utilizes a combination of energy, viscosity, and maximal regularity methods, including barrier constructions, semigroup theory, level-set methods, and measure-theoretic arguments, reflecting deep interplay between functional analysis, PDE, and applied mathematics.

7. Representative Table: Integral and Weighted Space Uniqueness Conditions

Operator Type	Uniqueness/Existence Class	Key Criterion/Bound	Reference
Linear, divergence	$u\in C^{2,1}\cap C$, boundary $a(x)\sim d(x)^{\gamma}$	Decay/integral condition (I_A/I_B) as $d(x)\to0$	(Nobili et al., 2020)
Degenerate $p$-Laplace	$u\in W^{1,2}_{loc}((0,T); L^2_v)$, $v,Q$ as weights	Minimal local integrability, weighted contraction	(Cruz-Uribe et al., 22 May 2024)
Weighted divergence	$u\in W^{1,p}_*(\Omega_T, \beta)$, $\beta\in$ Muckenhoupt	Weighted regularity + small-oscillation	(Fang et al., 22 Oct 2025)
Fully nonlinear	Bounded viscosity solutions, continuous up to $\partial\Omega$	Degeneracy implies implicit boundary	(Liu et al., 17 Jun 2024)
Spectral (deg. point)	Eigenfunction coeffs $(U_1^0(a)\neq0)$, initial data	Nonzero first Bessel mode for uniqueness	(Cannarsa et al., 2023)
Option pricing	$|u|\leq C \exp\{\cdots\}$, $V=o(1/v)$ as $v\to0$	Tikhonov–Täcklind class & sublinear growth	(Boyko, 14 Nov 2025)

These results synthesize a precise understanding of how degeneracy in parabolic dynamics impacts the uniqueness of solutions and underscores the critical role of weighted spaces, integral conditions, and boundary behavior in the admissible classes. The theory is an active research area, with ongoing refinement of sharpness, optimality, and applications across diverse modeling domains.