Regularity of Nonlinear Parabolic Differential Equations

Nonlinear parabolic differential equations describe a wide range of time-dependent diffusion processes in which the rate of change depends nonlinearly on the solution or its gradient. Prototypes include the porous medium equation, the p-Laplace heat flow and doubly nonlinear systems combining both degenerate and singular features. The study of regularity seeks to establish continuity, differentiability or Hölder continuity of weak solutions under minimal structural conditions. Achieving such regularity is crucial for ensuring uniqueness, stability and convergence of numerical approximations, and for deriving qualitative properties such as finite speed of propagation, interface regularity and asymptotic behaviour. Techniques centre on De Giorgi–Nash–Moser iteration, intrinsic scaling adapted to the local degeneracy or singularity, expansion of positivity, energy estimates and nonlinear potential theory. Boundary regularity introduces additional subtleties, requiring bespoke barrier constructions and refined measure-theoretical methods. Advances in this field not only unify diverse models under a common framework but also yield precise quantitative estimates for gradients and oscillations, with direct relevance to groundwater flow, heat conduction in heterogeneous media, phase transitions and image processing.

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Regularity of Nonlinear Parabolic Differential Equations publication trend

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Technical terms

Weak solution: A function satisfying the equation in an integral sense, allowing for lower regularity.

Degenerate/singular parabolic equation: A time-dependent PDE whose diffusion coefficient vanishes or blows up when the solution or its gradient approaches certain values.

Hölder continuity: A regularity property ensuring that the difference between solution values at two points is bounded by a constant times a fixed power of the distance.

Intrinsic scaling: A rescaling of space and time variables adapted to the local size of the solution or its gradient to restore uniform ellipticity or parabolicity.

De Giorgi iteration: A technique for improving integrability and oscillation bounds of a solution through successive energy estimates on shrinking cylinders.

Expansion of positivity: A principle that nonnegative solutions bounded below in a region propagate this positive lower bound to larger regions after a finite time.

Harnack inequality: A relation bounding the maximum of a nonnegative solution in a space-time cylinder by a constant times its minimum in a smaller cylinder, reflecting strong positivity propagation.

Modulus of continuity: A function quantifying the rate at which a solution can oscillate, with logarithmic type indicating slow growth of oscillation near the boundary.

References

1. Hölder regularity for degenerate parabolic double-phase equations. Journal of Differential Equations (2025).
2. On the Hölder regularity of signed solutions to a doubly nonlinear equation. Journal of Functional Analysis (2021).
3. On the Hölder regularity of signed solutions to a doubly nonlinear equation. Part II. Revista Matemática Iberoamericana (2022).
4. On the logarithmic type boundary modulus of continuity for the Stefan problem To the memory of Emmanuele DiBenedetto. Advances in Mathematics (2022).