Propagation dynamics of the monostable reaction-diffusion equation with a new free boundary condition (Q6587184)

Consider the following situation in a population modeling: Assume that a population of some species lives within the range given by \([g(t), h(t)]\). Assume also that it has a preferred density \(\delta\), so their habitat is not too crowded and not too isolated. To preserve this favored density, the population adjusts the boundaries of its habitat. In this paper, this type of condition is deduced as a free boundary condition and a free boundary problem is formulated to capture the evolution of the population. More precisely, the model describing such situation suggested in the paper reads, \N\[\Nu_t-du_{xx}= f(u), \quad t>0, \,\,\, g(t) <x<h(t),\N\]\N\[\Nu(t, g(t))=u(t,h(t))=\delta, \quad t>0,\N\]\N\[\Ng^{\prime}(t) = - \frac{d}{\delta} u_x (t, g(t)),\, \qquad t>0,\N\]\N\[\Nh^{\prime}(t) = - \frac{d}{\delta} u_x (t, h(t)), \,\qquad t>0,\N\]\N\[\N-g(0)=h(0)=h_0, \,\, u(0,x) =u_0(x), \quad\qquad -h_0\leq x\leq h_0, \N\]\Nwhere \(f\) is a monostable function and the initial conditions are from \N\[\N\{ \phi. \in C^2([-h_0,h_0])\,:\,\, \phi(x) >0 \text{ in } [-h_0,h_0], \,\,\, \phi (\pm h_0)=\delta \}.\N\]\NIn the paper, a set of results is obtained about the existence of the solutions and their asymptotic behavior.