H\"older estimates and large time behavior for a nonlocal doubly nonlinear evolution

Abstract

The nonlinear and nonlocal PDE $$|v_t|^{p-2}v_t+(-\Delta_p)^sv=0 \, ,$$ 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 H\"older estimates for viscosity solutions and relate the asymptotic behavior of solutions to the eigenvalue problem for the fractional $p$-Laplacian