Existence and Uniqueness of Solutions to Nonlinear Diffusion with Memory

TL;DR

This paper establishes the existence and uniqueness of weak solutions for a nonlinear diffusion equation with memory effects, using variational methods, energy estimates, and monotone operator theory.

Contribution

It introduces a rigorous analytical framework for nonlinear diffusion equations with memory, extending classical results to include convolution-type memory terms.

Findings

Existence of weak solutions under monotonicity and growth conditions

Uniqueness of solutions proved using energy estimates and monotone operator theory

Framework facilitates further study of memory-dependent diffusion processes

Abstract

This paper studies a nonlinear diffusion equation with memory: $$u_t=\nabla\cdot \big( D(x)\cdot\int_0^t K(t-s) \nabla\cdot\Phi(u(x,s))ds \big)+f(x,t)$$ Where $K$ is memory Kernel and $D(x)$ is bounded. Under monotonicity and growth conditions on $\Phi$, 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.