Spreading speeds and traveling waves for fractional equations with degenerate reactions
This paper focuses on studying the propagation phenomenon for fractional equations with degenerate reactions. Specifically, we prove the existence of a minimum traveling wave speed denoted by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c Superscript asterisk"> <mml:semantics> <mml:msup> <mml:mi>c</mml:mi> <mml:mo> ∗ </mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">c^*</mml:annotation> </mml:semantics> </mml:math> </inline-formula> so that traveling waves exist with any speed <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c greater-than-or-equal-to c Superscript asterisk"> <mml:semantics> <mml:mrow> <mml:mi>c</mml:mi> <mml:mo> ≥ </mml:mo> <mml:msup> <mml:mi>c</mml:mi> <mml:mo> ∗ </mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">c\geq c^*</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and non-existence of traveling waves with speed <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c greater-than c Superscript asterisk"> <mml:semantics> <mml:mrow> <mml:mi>c</mml:mi> <mml:mo>></mml:mo> <mml:msup> <mml:mi>c</mml:mi> <mml:mo> ∗ </mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">c>c^*</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Furthermore, we prove that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c Superscript asterisk"> <mml:semantics> <mml:msup> <mml:mi>c</mml:mi> <mml:mo> ∗ </mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">c^*</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is precisely the spreading speed and provides a Volpert-Volpert-Volpert type variational characterization for it in the homogeneous case. Our method is mainly based on monotone dynamic theory.
Authors
- Tao Zhou (ORCID: https://orcid.org/0000-0003-0561-2316)
- Lei Zhang (ORCID: https://orcid.org/0000-0003-2562-0698)
Institutions
- Anhui University (CN)
- Shaanxi Normal University (CN)
Publication Details
- Journal
- Proceedings of the American Mathematical Society
- Published
- 2025-05-14
- DOI
- https://doi.org/10.1090/proc/17282
- Primary Topic
- Fractional Differential Equations Solutions
- Type
- article
- Field-Weighted Citation Impact
- 0.00
Funders
- National Natural Science Foundation of China
- Fundamental Research Funds for the Central Universities