Remarks on parabolic equation with the conformable variable derivative in Hilbert scales

In this paper, we are interested in diffusion equations with conformable derivatives with variable order. We will study two different types of models: the initial value model and the nonlocal in time model. With different values of input values, we investigate the well-posedness of the mild solution in suitable spaces. We also prove the convergence of mild solution of the nonlocal problem to solutions of the initial problem. The main technique of our paper is to use the theory of Fourier series in combination with evaluation techniques for some generalized integrals. Our results are one of the first directions on the diffusion equation with conformable variable derivative in Hilbert scales.