Source integrability and the frontier α=1/2 in time-fractional diffusion
Abstract
Let Ω⊂ℝd be a bounded Lipschitz domain, let α∈(0,1), and let ∂tα denote the Caputo derivative. For ∂tαu−∇·(A(x)∇ u)=f with homogeneous Dirichlet condition and initial datum g∈ L2(Ω), we identify which of the thresholds appearing in the energy method belong to the method and which to the equation. We compare f∈ L2((0,T);L2(Ω)) and f∈ Lr((0,T);L2(Ω)) with r>2/α; in the subdiffusive range the second implies the first and not conversely. Existence is not at issue and we claim none of it: it is known, for both hypotheses and under weaker ones, from the work of Zacher. What we determine is the exact set of exponents each route admits and what the exponents left outside it cost. Under the weaker hypothesis the construction below reaches α∈(1/2,1), under the stronger one every α∈(0,1), and in both cases the solution lies in L∞((0,T);L2(Ω)), by the mode representation and not by the energy route, which yields no such bound under the weaker hypothesis. The restriction α>1/2 comes from the compactness criterion, whose admissible exponents form [1,2α); the exponent r>2/α is exactly what the Grönwall step consumes. A concentrating family shows that nothing weaker suffices along that route, and the mode representation lowers the requirement to 1/α. The frontier α=1/2 belongs to the equation: for every α∈(0,1/2] there is f∈ L2((0,T);L2(Ω)) whose weak solution leaves L∞((0,T);L2(Ω)). Uniqueness and Hadamard well-posedness follow.
Related
- Inverse source problem of variable-exponent time-fractional diffusion equations
- Inverse Source Problem for a Time-Fractional Diffusion-Wave Equation with a Singular Inverse-Square Potential
- Subdivision-based collocation methods for time fractional convection diffusion reaction equation with environmental and biomedical applications
- Recovering the Fractional Order and an Initial Value Simultaneously in a Time-Fractional Diffusion-Wave Equation
- A coercive space-time variational approach to fractional diffusion problems
BibTeX
@misc{vasquezmartinez2026source,
title = {{Source integrability and the frontier $α=\tfrac12$ in time-fractional diffusion}},
author = {Edison A. Vásquez-Martínez and Mauricio A. Londoño-Arboleda},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.30377},
eprint = {2610.30377},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.30377},
}
RIS
TY - UNPB
TI - Source integrability and the frontier α=1/2 in time-fractional diffusion
AU - Edison A. Vásquez-Martínez
AU - Mauricio A. Londoño-Arboleda
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.30377
UR - https://arxiv.org/abs/2610.30377
AB - Let Ω⊂ℝ^d be a bounded Lipschitz domain, let α∈(0,1), and let ∂_t^α denote the Caputo derivative. For ∂_t^αu−∇·(A(x)∇ u)=f with homogeneous Dirichlet condition and initial datum g∈ L^2(Ω), we identify which of the thresholds appearing in the energy method belong to the method and which to the equation. We compare f∈ L^2((0,T);L^2(Ω)) and f∈ L^r((0,T);L^2(Ω)) with r>2/α; in the subdiffusive range the second implies the first and not conversely. Existence is not at issue and we claim none of it: it is known, for both hypotheses and under weaker ones, from the work of Zacher. What we determine is the exact set of exponents each route admits and what the exponents left outside it cost. Under the weaker hypothesis the construction below reaches α∈(1/2,1), under the stronger one every α∈(0,1), and in both cases the solution lies in L^∞((0,T);L^2(Ω)), by the mode representation and not by the energy route, which yields no such bound under the weaker hypothesis. The restriction α>1/2 comes from the compactness criterion, whose admissible exponents form [1,2α); the exponent r>2/α is exactly what the Grönwall step consumes. A concentrating family shows that nothing weaker suffices along that route, and the mode representation lowers the requirement to 1/α. The frontier α=1/2 belongs to the equation: for every α∈(0,1/2] there is f∈ L^2((0,T);L^2(Ω)) whose weak solution leaves L^∞((0,T);L^2(Ω)). Uniqueness and Hadamard well-posedness follow.
ER -