Spreading and vanishing in a free boundary problem for nonlinear diffusion equations with a given forced moving boundary
The paper is concerned with the study of the free boundary problem \(u_t=u_{xx}+f(u)\), \(t>0\), \(0<x<h(t)\) subject to the following conditions: \(u(t,ct)=u(t,h(t))=0\), \(h'(t)=-\mu u_x(t, h(t))\) for \(t>0\) and \(h(0)=h_0\), \(u(0,x)=u_0(x)\), \(0\leq x\leq h_0\). Here \(c,\mu, h_0>0\) are given constants and \(f\in C^1\) satisfies \(f(0)=f(1)=0\), \(f'(1)<0\), \(f(u)<0\) for \(u>1\). The dynamical behavior of solutions to the above problem is considered in the frame of three type on extra-assumptions on \(f\): monostable case; bistable case; combustion case. It is derived that the unique solution exhibits exactly one of the behaviors: spreading, vanishing and transition. The asymptotic profile of the solution over the whole domain when spreading occurs.
scientific article; zbMATH DE number 6864107
scientific article