2024/12/04 by Takashi Kumagai, Jian Wang, Kumagai, Takashi +3
Mathematics · Computer Science · #Nonlinear Partial Differential Equations #Advanced Mathematical Modeling in Engineering #Differential Equations and Boundary Problems
paper · pdf · doi:10.48550/arxiv.2412.03770
In this paper, we study the local boundedness of local weak solutions to the following parabolic equation associated with fractional p-Laplacian type operators ∂t u(t,x)-p.v.∫\Rd|u(t,y)-u(t,x)|p-2(u(t,y)-u(t,x))J(t;x,y) dy=0, (t,x)∈ \R× \Rd, where p.v. means the integral in the principal value sense, p∈(1,∞) and J(t;x,y) is comparable to the kernel of the fractional p-Laplacian operator |x-y|-d-sp with s∈(0,1) and uniformly in (t;x,y)∈\R×\Rd×\Rd. Unlike existing results in the literature, the local boundedness of the solutions obtained in this paper extends the known results for the linear case (i.e., the case that p=2), in particular with a nonlocal parabolic tail that uses the L1-norm in time for all p∈ (1,∞). The proof is based on a new level set truncation in the De Giorgi-Nash-Moser iteration and a careful choice of iteration orders, as well as a general Caccioppoli-type inequality that is efficiently applied to fractional p-Laplacian type operators with all p>1.