Local boundedness of solutions to non-local parabolic equations modeled on the fractional p-Laplacian

Abstract

We state and prove estimates for the local boundedness of subsolutions of non-local, possibly degenerate, parabolic integro-differential equations of the formwhere P.V. means in the principle value sense, p ∈ (1, ∞) and the kernel obeys K(x, y, t) ≈ |x − y| n+ps for some s ∈ (0, 1), uniformly in (x, y, t) ∈ R n × R n × R.

Harnack's inequality for parabolic nonlocal equations

“…The proof is valid for p > 2 and not p = 2 that is considered in this paper. The bounds established in [18] depend on the supremum-version of the tail (4). In that sense they are weaker than those established in the present paper.…”

“…In [18] by the author, local boundedness of solutions to degenerate nonlocal parabolic equations of p−Laplace type is proved. The proof is valid for p > 2 and not p = 2 that is considered in this paper.…”

Continuity of solutions to a nonlinear fractional diffusion equation

“…In recent years there has been a surge of interest around the operator (1.2), after its introduction in [20]. In particular, equation (1.1) has been studied in [1,25,26,31,33,34] and [35]. References [25,26,33] and [34] dealt with existence and uniqueness of solutions, together with their long time asymptotic behaviour.…”

“…In [35], some regularity of the semigroup operator generated by (− p ) s was studied. In [31], the local boundedness of weak solutions of (1.1) is proved.…”

Nonlocal diffusion equations with dynamical boundary conditions

“…In [8] the authors deal with nonlocal fractional problems that approximate Steklov eigenvalues. For extra references concerning evolution problems involving the fractional p−Laplacian we quote [1,18,19,20,22] and references therein.…”