Existence and Uniqueness of Solutions to Nonlinear Diffusion with Memory

2025/07/22 by Chen, Yixian
#Analysis of PDEs (math.AP) #FOS: Mathematics #[2020] 35K55 35B45 35D30 35R60

paper · doi:10.48550/arxiv.2507.16659

Abstract

This paper studies a nonlinear diffusion equation with memory: u_t=∇⋅ ( D(x)⋅∫₀^t K(t-s) ∇⋅Φ(u(x,s))ds )+f(x,t) Where K is memory Kernel and D(x) is bounded. 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 basis for studying memory-type diffusion.

Citations

• Optimal decay estimates for time-fractional and other non-local subdiffusion equations via energy methods
• A consistent Fubini-Tonelli theorem for nonmeasurable functions

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