Maximal $$L_p$$-regularity for x-dependent fractional heat equations with Dirichlet conditions
Year: 2024
Type: article
Abstract: We prove optimal regularity results in $$L_p$$ L p -based function spaces in space and time for a large class of linear parabolic equations with a nonlocal elliptic operator in bounded domains with limited smoothness. Here the nonlocal operator is given by a strongly elliptic and even pseudodifferential operator P of order 2 a ( $$0 0 < a < 1 ) with nonsmooth x -dependent coefficients. This includes the prominent case of the fractional Laplacian $$(-\Delta )^a$$ ( - Δ ) a , as well as elliptic operators $$(-\nabla \cdot A(x)\nabla +b(x))^a$$ ( - ∇ · A ( x ) ∇ + b ( x ) ) a . The proofs are based on general results on maximal $$L_p$$ L p -regularity and its relation to $$\mathcal {R}$$ R -boundedness of the resolvent of the associated (elliptic) operator. Finally, we apply these results to show existence of strong solutions locally in time for a class of nonlinear nonlocal parabolic equations, which include a fractional nonlinear diffusion equation and a fractional porous medium equation after a transformation. The nonlinear results are new in the case of domains with boundary; the linear results are so when P is x -dependent nonsymmetric.
Source: Mathematische Annalen
Authors: Helmut Abels, Gerd Grubb
Institutions: University of Regensburg, University of Copenhagen
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $2,990
Funder: Københavns Universitet
OpenAlex ID: W4402974395