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In this paper, we discuss the optimality of the assumptions used, in a previous paper, to prove existence and symmetry of minimizers of the fractional constrained variational problem: where c is a prescribed number. More precisely, we will show that if one of the conditions, used to prove that all minimizers of the above constrained variational problem, are radial and radially decreasing for all c, do not hold true, then there are several interesting situations: In the fractional setting, such a study is more subtle than in the classical one. We take advantage of some brilliant results obtained recently in Cabre and Sire (Anal. PDEs, ), Dyda (Fractional calculus for power functions, ) and Hajaiej et al. (Necessary and sufficient conditions for the fractional Gagliardo-Nirenberg inequalities and applications to Navier-Stokes and generalized Boson equations, ). [ABSTRACT FROM AUTHOR]

This paper deals with mathematical problems related to space-dependent anomalous diffusion processes. Namely, we investigate diffusion equations with time-fractional derivatives of space-dependent variable order. We establish that variable-order time-fractional Cauchy problems admit a unique weak solution and prove that the space-dependent variable-order coefficient is uniquely determined by the knowledge of a suitable time sequence of partial initial-boundary maps. [ABSTRACT FROM AUTHOR]