Hölder estimates and large time behavior for a nonlocal doubly nonlinear evolution

Abstract

The nonlinear and nonlocal PDE $$|v_t{|}^{p-2}{v}_t+(-{\Delta }_p{)}^{s}v=0,$$ where $$(-{\Delta }_p{)}^{s}v(x,t)=2\text{PV}{\int }_{{\mathbb{R}}^n}\frac{|v(x,t)-v(x+y,t){|}^{p-2}(v(x,t)-v(x+y,t))}{|y{|}^{n+sp}}dy,$$ has the interesting feature that an associated Rayleigh quotient is non-increasing in time along solutions. We prove the existence of a weak solution of the corresponding initial value problem which is also unique as a viscosity solution. Moreover, we provide Hölder estimates for viscosity solutions and relate the asymptotic behavior of solutions to the eigenvalue problem for the fractional $p$-Laplacian.

References in corpus (3)

• Local behavior of fractional $p$-minimizers
• Nonlocal equations with measure data
• The first non-zero Neumann $p-$fractional eigenvalue