Existence and Uniqueness of Solutions to Nonlinear Diffusion with Memory
Yixian Chen
TL;DR
The paper analyzes a nonlinear diffusion equation with memory encoded by a convolution in time: $u_t=\nabla\cdot\big( D(x)\int_0^t K(t-s)\nabla\Phi(u)\,ds\big)+f$, on a bounded Lipschitz domain. Through Galerkin (orthogonal) approximations, energy estimates, and monotone operator methods within a variational framework, it establishes existence and uniqueness of weak solutions in $L^2(0,T;H_0^1(\Omega))$, with uniform bounds that enable compactness and passage to the limit. The memory term is treated as a bounded convolution operator, and weak convergence arguments together with continuity of $\Phi'$ identify the limit as a weak solution of the original problem. The results provide a rigorous foundation for memory-type diffusion models and pave the way for extensions to more general kernels, nonlinearities, and stochastic or homogenization contexts.
Abstract
This paper studies a nonlinear diffusion equation with memory: $$u_t=\nabla\cdot \big( D(x)\cdot\int_0^t K(t-s) \nabla\cdotΦ(u(x,s))ds \big)+f(x,t)$$ 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.