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Spreading in a shifting environment modeled by the diffusive logistic equation with a free boundary. (English) Zbl 1408.35227

The authors study the long-time behavior of the unique (classical) solution \((u,h)\) to the Cauchy free boundary problem of the diffusive logistic equation \(u_t=du_{xx}+A(x-ct)u-bu^2\) in \( D= \{ (x,t)\in \mathbb{R}^2_+: x < h(t) \} \). Here, \(u\) represents the population density of the concerned species and \(h\) represents the moving boundary to be determined, while \(c>0\) denotes the environment shifting speed and \(A\) is a Lipschitz continuous function, such that is strictly increasing over \(]-l_0,0[\) (\(l_0>0\)) and constant otherwise, in order to describe the shifting environment. At the fixed boundary \(x=0\), a no-flux boundary condition is assumed, while, at the free boundary, \(h'(t)=-\mu u_x(t,h(t))\) is stated. The constants \(d,b,\mu\) are assumed to be positive. The authors prove that, when \(c\geq c_0\), the species always dies out in the long-run, and, when \(0< c< c_0\), the long-time behavior of the species is determined by a trichotomy in the form (a) vanishing, (b) borderline spreading, or (c) spreading. To see which case, in the trichotomy, happens with a given initial population \(u_0\), the authors take \(u_0=\sigma \phi\), where \(\sigma\) is a positive parameter and \(\phi\in C^2([0,h_0])\) is such that \(\phi>0\) in \([0,h_0[\), \(\phi'(0)=\phi(h_0)=0\) and \(\phi'(h_0)<0\). Then, they show that there exists a critical \(\sigma_0\in ]0,+\infty]\) such that vanishing happens for \(\sigma< \sigma_0\), borderline spreading happens for \(\sigma=\sigma_0\), and spreading happens for \(\sigma>\sigma_0\). The proofs rely on comparison principles by taking the contradiction argument into account.

MSC:

35R35 Free boundary problems for PDEs
35K20 Initial-boundary value problems for second-order parabolic equations
92B05 General biology and biomathematics
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