Noncompact Time–Space Fractional Diffusion with Volterra Memory

We investigate a class of nonlinear time–space fractional reaction–diffusion equations with Volterra memory on the whole space RN. The temporal dynamics are governed by a Caputo derivative of order 0<α<1, while spatial diffusion is described by the fractional Laplacian (−Δ)s, 0<s<1. A central difficulty is that the associated translation-invariant fractional solution family is noncompact on L2(RN), so standard compactness-based fixed-point arguments are not directly applicable. To overcome this obstruction, we develop a measure-of-noncompactness estimate for the complete mild-solution mapping. The estimate simultaneously accounts for the singular time-fractional convolution and the nested Volterra memory term. An exponentially weighted argument is then used to verify the countable-set compactness condition required by Mönch’s continuation principle without imposing a smallness restriction on the relevant coefficients. Combined with an a priori estimate, this yields the existence of at least one mild solution in C([0,T];L2(RN)) under Carathéodory, growth, and measure-of-noncompactness assumptions. The abstract result is applied to a class of nonlinear fractional reaction–diffusion equations with bounded measurable coefficients. A fully specified one-dimensional problem with fractional orders α=3/4 and s=1/2 and an exponentially decaying memory kernel is also presented, for which the constants entering the existence framework can be evaluated explicitly.