Existence and Uniqueness of Solutions to Nonlinear Diffusion with Memory

This paper studies a nonlinear diffusion equation with memory: $$u_t=\mathrm{\nabla }\cdot (D(x)\cdot {\int }_0^{t}K(t-s)\mathrm{\nabla }\cdot \mathrm{\Phi }(u(x,s))ds)+f(x,t)$$ Where $K$ is memory Kernel and $D(x)$ is bounded. Under monotonicity and growth conditions on $\mathrm{\Phi }$, 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.