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 |
| Published: |
Hindawi Limited
2010-01-01
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| Series: | International Journal of Differential Equations |
| Online Access: | http://dx.doi.org/10.1155/2010/315421 |
| Summary: | 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. |
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| ISSN: | 1687-9643 1687-9651 |