Large time behavior for a nonlocal nonlinear gradient flow

Abstract

<p style='text-indent:20px;'>We study the large time behavior of the nonlinear and nonlocal equation</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ v_t+(- \Delta_p)^sv = f \, , $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ p\in (1, 2)\cup (2, \infty) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ s\in (0, 1) $\end{document}</tex-math></inline-formula> and</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ (- \Delta_p)^s v\, (x, t) = 2 \, {\rm{P.V.}} \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. $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>This equation arises as a gradient flow in fractional Sobolev spaces. We obtain sharp decay estimates as <inline-formula><tex-math id="M3">\begin{document}$ t\to\infty $\end{document}</tex-math></inline-formula>. The proofs are based on an iteration method in the spirit of J. Moser previously used by P. Juutinen and P. Lindqvist.</p>