On the Speed of Spread for Fractional Reaction-Diffusion Equations
The fractional reaction diffusion equation 𝜕𝑡𝑢+𝐴𝑢=𝑔(𝑢) is discussed, where 𝐴 is a fractional differential operator on ℝ of order 𝛼∈(0,2), the 𝐶1 function 𝑔 vanishes at 𝜁=0 and 𝜁=1, and either 𝑔≥0 on (0,1) or 𝑔<0 near 𝜁=0. In the case of nonnegative g, it is shown that solutions with initial suppo...
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| Format: | Article |
| Language: | English |
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Wiley
2010-01-01
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| Series: | International Journal of Differential Equations |
| Online Access: | http://dx.doi.org/10.1155/2010/315421 |
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| author | Hans Engler |
| author_facet | Hans Engler |
| author_sort | Hans Engler |
| collection | DOAJ |
| description | The fractional reaction diffusion equation 𝜕𝑡𝑢+𝐴𝑢=𝑔(𝑢)
is discussed, where 𝐴 is a fractional differential operator on ℝ of order
𝛼∈(0,2), the 𝐶1 function 𝑔 vanishes at 𝜁=0 and 𝜁=1, and either
𝑔≥0 on (0,1) or 𝑔<0 near 𝜁=0. In the case of nonnegative g,
it is shown that solutions with initial support on the positive half axis
spread into the left half axis with unbounded speed if 𝑔(𝜁) satisfies some
weak growth condition near 𝜁=0 in the case 𝛼>1, or if 𝑔 is merely
positive on a sufficiently large interval near 𝜁=1 in the case 𝛼<1. On the other hand, it shown that solutions spread with finite speed if
𝑔(0)<0. The proofs use comparison arguments and a suitable family
of travelling wave solutions. |
| format | Article |
| id | doaj-art-30a1224b3c0f47b58748b7248798efb3 |
| institution | Kabale University |
| issn | 1687-9643 1687-9651 |
| language | English |
| publishDate | 2010-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Differential Equations |
| spelling | doaj-art-30a1224b3c0f47b58748b7248798efb32025-02-03T06:48:28ZengWileyInternational Journal of Differential Equations1687-96431687-96512010-01-01201010.1155/2010/315421315421On the Speed of Spread for Fractional Reaction-Diffusion EquationsHans Engler0Department of Mathematics, Georgetown University, Box 571233, Washington, DC 20057, USAThe fractional reaction diffusion equation 𝜕𝑡𝑢+𝐴𝑢=𝑔(𝑢) is discussed, where 𝐴 is a fractional differential operator on ℝ of order 𝛼∈(0,2), the 𝐶1 function 𝑔 vanishes at 𝜁=0 and 𝜁=1, and either 𝑔≥0 on (0,1) or 𝑔<0 near 𝜁=0. In the case of nonnegative g, it is shown that solutions with initial support on the positive half axis spread into the left half axis with unbounded speed if 𝑔(𝜁) satisfies some weak growth condition near 𝜁=0 in the case 𝛼>1, or if 𝑔 is merely positive on a sufficiently large interval near 𝜁=1 in the case 𝛼<1. On the other hand, it shown that solutions spread with finite speed if 𝑔(0)<0. The proofs use comparison arguments and a suitable family of travelling wave solutions.http://dx.doi.org/10.1155/2010/315421 |
| spellingShingle | Hans Engler On the Speed of Spread for Fractional Reaction-Diffusion Equations International Journal of Differential Equations |
| title | On the Speed of Spread for Fractional Reaction-Diffusion Equations |
| title_full | On the Speed of Spread for Fractional Reaction-Diffusion Equations |
| title_fullStr | On the Speed of Spread for Fractional Reaction-Diffusion Equations |
| title_full_unstemmed | On the Speed of Spread for Fractional Reaction-Diffusion Equations |
| title_short | On the Speed of Spread for Fractional Reaction-Diffusion Equations |
| title_sort | on the speed of spread for fractional reaction diffusion equations |
| url | http://dx.doi.org/10.1155/2010/315421 |
| work_keys_str_mv | AT hansengler onthespeedofspreadforfractionalreactiondiffusionequations |