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A nonlinear age-dependent model with spatial diffusion. (English) Zbl 1116.35062

The authors have studied the existence and uniqueness of a positive solution to the following problem \[ \begin{cases} u_a-\Delta u+q(x,a)u=g(x,a,u) \quad &\text{in } \Omega\times(0,A)\\ u=0\quad &\text{on }\partial \Omega\times(0,A)\\ u(x,0)= \int^\Delta_0\beta(x,a)u(x,a)da\quad & \text{in }\Omega,\end{cases} \] where \(\Omega \subset\mathbb{R}^H\) is a bounded domain, \(A>0\), \(\beta\) is a bounded, nonnegative function, \(q\) and \(g\) are measurable functions. This problem describes the dynamics of an age-structured population. The properties of an eigenvalue problem related to the equation as well as the sub-supersolution method, are used.

MSC:

35K57 Reaction-diffusion equations
35Q80 Applications of PDE in areas other than physics (MSC2000)
92D25 Population dynamics (general)
35K20 Initial-boundary value problems for second-order parabolic equations
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References:

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