## 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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