Nonlinear Diffusion Equations and Parabolic Problems

Nonlinear diffusion equations encompass a broad class of time-dependent partial differential equations in which the diffusive flux depends nonlinearly on the solution or its gradient. These models generalise the classical heat equation to account for phenomena such as degenerate or singular diffusion, anisotropic transport, memory effects and spatial heterogeneity. Parabolic problems of this kind arise in porous-media flow, phase transitions, biological pattern formation, image processing and materials science. Key analytical challenges include establishing existence, uniqueness and regularity of solutions, understanding finite-time blow-up or extinction, and characterising asymptotic behaviour. Fractional-order operators extend the framework to anomalous diffusion with long-range interactions, while variable-exponent and pseudo-parabolic formulations capture media with evolving or history-dependent properties. Advances in this field continue to blend functional analysis, variational methods and geometric insights to reveal intricate interplay between nonlinearity, nonlocality and time evolution.

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Research from all publishers

Recent work on parabolic equations with nonstandard growth has demonstrated optimal global second-order regularity and improved gradient integrability for solutions subject to inhomogeneous sources in spatially and temporally varying settings. These results extend classical Calderón-Zygmund theory to variable-exponent frameworks and yield sharper continuity and compactness properties. In the realm of nonlocal diffusion, semilinear Kirchhoff-type models driven by fractional Laplacian operators with logarithmic nonlinearities have been shown to support global weak solutions alongside criteria for finite-time blow-up; precise upper and lower bounds on blow-up times have been obtained by combining potential-well and concavity methods. Further progress in pseudo-parabolic problems with p(x)-Laplacian operators and memory terms has elucidated how initial energy levels control long-time dynamics, delivering unified criteria for global existence, decay rates and blow-up phenomena in systems with spatial heterogeneity and damping effects.

Nonlinear Diffusion Equations and Parabolic Problems publication trend

The graph below shows the total number of articles in nonlinear diffusion equations and parabolic problems across all publications each year (not limited to Nature Index journals).

Technical terms

Nonlinear diffusion: A process in which the flux depends nonlinearly on the solution or its gradient, modelled by ut = div(D(u,∇u)).

Parabolic equation: A time-dependent PDE characterised by smoothing behaviour, typified by the heat equation ut = Δu.

Fractional Laplacian: A nonlocal operator (−Δ)^s, 0

Kirchhoff-type problem: A model where the diffusion coefficient depends on global norms of the solution, reflecting tension or memory effects.

Variable exponent p(x,t): A generalisation of the p-Laplacian where the integrability exponent varies in space and time, leading to nonstandard growth conditions.

Pseudo-parabolic equation: A PDE combining parabolic and additional memory or damping terms, often of the form ut − Δut + Δu = f.

References

1. Optimal global second-order regularity and improved integrability for parabolic equations with variable growth. Advances in Nonlinear Analysis (2024).
2. Global Existence, Blowup, and Asymptotic Behavior for a Kirchhoff-Type Parabolic Problem Involving the Fractional Laplacian with Logarithmic Term. Mathematics (2023).
3. Lifespan of solutions for a class of pseudo-parabolic equation with weak-memory. Alexandria Engineering Journal (2020).
4. Global Existence and Blow-Up for the Pseudo-parabolic p(x)-Laplacian Equation with Logarithmic Nonlinearity. Journal of Nonlinear Mathematical Physics (2021).