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Long range diffusion reaction model on population dynamics. (English) Zbl 0976.35032

The author considers the initial value problem \((*) \;\frac{\partial S}{\partial t} - C \Delta^{(2)} S=aS^n, \;S(x,0)=g(x), \;x \in \mathbb R^2\), where \(\Delta^{(2)} = \sum^2_{i,j=1} \frac{\partial^4}{\partial x_i^2 \partial x^2_j}, \;S \in \mathbb R, \;a\) and \(C\) are some constants. He proves existence and uniqueness of the initial value problem \((*)\) in \(L^{p,q}, \;p =\frac{3}{2} (n-1), \;q=\frac{1}{2}(n-1)\) for \(n>3\) whenever \(g(*)\) is sufficiently small in the norm of \(L^{p,q}.\)

MSC:

35K30 Initial value problems for higher-order parabolic equations
92D25 Population dynamics (general)
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