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http://www.mhikari.com/imfpassword2009/17202009/awawdehIMF17202009.pdf
So notice that the condition where standard method using a Banach fixed point theorem solves the integral equation is when F is Lipschitz continuous meaning
This sort of thing is FALSE for nonlinearities like KPP and possibly for others. In other words, the standard technique for proving that a solution to the nonlinear fractional diffusion equation with general nonlinear F fails. Therefore although this does not prove that instability must occur in the solution, it does gives us some further intuition that beyond the KPP instability for nonLipschitz F, especially when iterations of F produces a complicated Julia set would produce instability/turbulence, and this phenomenon may be provably wider than the case of KPP F=u^2u. I’m led to the conjecture that whenever F has a complicated and large Julia set, instability would be expected in the solution (even without any probabilistic interpretation such as branching process).
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