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September 23, 2025Open Access

Existence and Uniqueness of Solutions to Nonlinear Diffusion with Memory in Random Media

Key Points

The study establishes the existence and uniqueness of weak solutions to nonlinear diffusion equations in random media, expanding on previous theories.
A key finding is the rigorous result achieved under conditions of monotonicity and growth, facilitating future research in memory-type diffusion.
The analysis incorporates methodologies like orthogonal approximation and monotone operator theory, contributing to the understanding of the complex diffusion dynamics.
This research highlights the importance of variational frameworks in addressing the convolution structure related to memory in diffusion processes.

Abstract

This paper studies a nonlinear diffusion equation with memory and spatial randomness: uₜ= (D (x, w) ₀ᵗ K (t-s) Φ (u (x, s) ) ds) +f (x, t) where K is memory Kernel and D (x, w) is bounded random coefficient. Under monotonicity and growth conditions on Φ, the existence and uniqueness of weak solution is established. The analysis employs Orthogonal approximation, energy estimates, and monotone operator theory. The convolution structure is handled within variational frameworks. The result provides a rigorous basis for studying memory-type diffusion in heterogeneous media.

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