Energy methods and optimal decay for time-fractional equations
Abstract
We study energy methods for the time-fractional equation ∂tα(u−u0)−(a(t,x)ux)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∂xF≥0, σQ≤0, and the data are in L1∩ L∞. Using Yosida approximations of the kernel, an iteration on the L2m norms gives the parabolic decay |u(t)|L∞≤ C|u0|L1t−α/2. In L2 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 ∂tα(u−u0)−uxx+(|u|q−1u)x=0, q>1, the parabolic rate is optimal for q≥2, and for 1<q<2 the convection improves it. An L1 time-regularity estimate of Bénilan--Crandall type gives the pointwise flux bound |u|q−1u−ux=O(|u0|L1t−α). This bound takes the place of the Ole\uınik estimate and yields the optimal rate |u(t)|L∞≲ t−α/q for 1<q<2. For nonnegative data, matching lower bounds hold for the Lp norms.
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BibTeX
@misc{ignat2026energy,
title = {{Energy methods and optimal decay for time-fractional equations}},
author = {Liviu I. Ignat},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.07185},
eprint = {2611.07185},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.07185},
}
RIS
TY - UNPB
TI - Energy methods and optimal decay for time-fractional equations
AU - Liviu I. Ignat
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.07185
UR - https://arxiv.org/abs/2611.07185
AB - We study energy methods for the time-fractional equation ∂_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∂_xF≥0, σQ≤0, and the data are in L^1∩ L^∞. Using Yosida approximations of the kernel, an iteration on the L^2^m norms gives the parabolic decay |u(t)|_L^∞≤ C|u_0|_L^1t^−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 ∂_t^α(u−u_0)−u_xx+(|u|^q−1u)_x=0, q>1, the parabolic rate is optimal for q≥2, 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−1u−u_x=O(|u_0|_L^1t^−α). This bound takes the place of the Ole\uınik estimate and yields the optimal rate |u(t)|_L^∞≲ t^−α/q for 1<q<2. For nonnegative data, matching lower bounds hold for the L^p norms.
ER -