Energy methods and optimal decay for time-fractional equations

Abstract

We study energy methods for the time-fractional equation $\partial_t^α(u-u_0)-(a(t,x)u_x)_x+(F(t,x,u))_x=Q(t,x,u)$ on the real line, where $0<α<1$, the coefficient $a$ is measurable and bounded from above and from below by positive constants, $σ\partial_xF\ge0$, $σQ\le0$, and the data are in $L^1\cap L^\infty$. Using Yosida approximations of the kernel, an iteration on the $L^{2^m}$ norms gives the parabolic decay $\|u(t)\|_{L^\infty}\le C\|u_0\|_{L^1}t^{-α/2}$. In $L^2$ it gives the rate $t^{-α/4}$, which, already for the linear equation, improves the rate $t^{-α/5}$ obtained before by the classical energy method. For the convection--diffusion equation $\partial_t^α(u-u_0)-u_{xx}+(|u|^{q-1}u)_x=0$, $q>1$, the parabolic rate is optimal for $q\ge2$, and for $1<q<2$ the convection improves it. An $L^1$ time-regularity estimate of Bénilan--Crandall type gives the pointwise flux bound $|u|^{q-1}u-u_x=O(\|u_0\|_{L^1}t^{-α})$. This bound takes the place of the Ole\uınik estimate and yields the optimal rate $\|u(t)\|_{L^\infty}\lesssim t^{-α/q}$ for $1<q<2$. For nonnegative data, matching lower bounds hold for the $L^p$ norms.