Continuity of solutions to a nonlinear fractional diffusion equation

Abstract

We study a parabolic equation for the fractional p-Laplacian of order s, for $$p\ge 2$$ p ≥ 2 and $$0<s<1$$ 0 < s < 1 . We provide space-time Hölder estimates for weak solutions, with explicit exponents. The proofs are based on iterated discrete differentiation of the equation in the spirit of Moser’s technique.

Existence of variational solutions to doubly nonlinear nonlocal evolution equations via minimizing movements

“…The comparision map v in (4. 16) is chosen to satisfy v ∈ L p (0, T ; W s,p u 0 (Ω)) with ∂ t v ∈ L φ (Ω T ) and v(0) ∈ L φ (Ω). Moreover, we extend v to negative times t < 0 by v(t) = v(0).…”

“…Local boundedness and Hölder regularity has been proved in [33]. Explicit Hölder regularity has been obtained in [16]. The latter two authors have proved local boundedness estimates for the double phase nonlocal parabolic equation in [57].…”

Local boundedness of variational solutions to nonlocal double phase parabolic equations

“…Other relevant papers include [2,3,4,18] The regularity theory of p, q growth problems was started by Marcellini in a series of novel papers [20,21,22,23]. There is a large body of work dealing with problems of (p, q)-growth as well as other nonstandard growth problems, for which we point to the surveys in [24,25].…”

Existence of variational solutions to nonlocal evolution equations via convex minimization

“…Local boundedness and Hölder regularity has been proved in [22]. Explicit Hölder regularity has been obtained in [7].…”