2014
DOI: 10.1007/978-1-4471-5526-3_9
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Abstract: We consider the following shadow system of the Gierer-Meinhardt system with saturation: ⎧ ⎪ ⎨ ⎪ ⎩ A t = 2 ∆A − A + A 2 ξ(1+kA 2) in Ω × (0, ∞), τ ξ t = −ξ + 1 |Ω| Ω A 2 dx in (0, +∞), ∂A ∂ν = 0 on ∂Ω × (0, ∞), where > 0 is a small parameter, τ ≥ 0, k > 0 and Ω ⊂ R n is smooth bounded domain. The case k = 0 has been studied by many authors in recent years. Here we give some sufficient conditions on k for the existence and stability of stable spiky solutions. In the one-dimensional case we have a complete answer…

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