Abstract

In this paper, we study some qualitative properties related to Semi-linear Volterra Diffusion Equations on networks $$u_t(x,t)=\Delta_{\omega}u(x,t)+\int_0^t k(t-\tau)|u|^{p-1}(x,\tau)u(x,\tau)d\tau$$ where $\Delta _{\omega}$ is called the discrete weighted Laplacian operator. Under some appropriate hypotheses, we prove the existence and uniqueness of the local solution via Banach fixed point theorem.  Besides, we provide several types of the maximum principles to this equation. We also show that the solution of the problem blow-up in a finite time. Finally, we verify our results through some numerical examples.

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