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In this manuscript, we will study the asymptotic behavior for a class ofnonlocal diffusion equations associated with the weighted fractional$\wp(\cdot)-$Laplacian operator involving constant/variable exponent. In thecase of constant exponents, under some appropriate conditions, we will studythe existence of solutions and asymptotic behavior of solutions by employingthe subdifferential approach and we will study the problem when $\wp$ goes to$\infty$. Already, for case the weighted fractional $\wp(\cdot)$-Laplacianoperator, we will also study the asymptotic behavior of the problem solutionwhen $\wp(\cdot)$ goes to $\infty$, in the whole or in a subset of the domain(the problem involving the fractional $\wp(\cdot)$-Laplacian presents adiscontinuous exponent). To obtain the results of the asymptotic behavior inboth problems it will be via Mosco convergence.

On asymptotic behavior for a class of diffusion equations involving the fractional $\wp(\cdot)-$Laplacian as $\wp(\cdot)$ goes to $\infty$