Local stability of traveling-wave solutions of nonlinear reaction-diffusion equations
The authors investigate local stability of traveling-wave solutions for nonlinear reaction-diffusion equations in various weighted Banach spaces. The problems under consideration are of the form \[ u_t=u_{xx}+f(u) \] for \(t>0\), \(x\in{\mathbb R}\) subject to the condition \(u(x,0)=\bar u(x)\), \(x\in{\mathbb R}\). The nonlinearity \(f\) is \(C^1\) on \([0,1]\) and \(C^{2,\alpha}\) on \((0,1)\) with bounded second derivative, and satisfies \(f(0)=f(1)=0\), \(f'(0)=0\), \(f'(1)<0\), \(f'(s)>0\), \(f'(1-s)<0\) for small \(s>0\), \(f(x)\geq 0\) on \([0,1]\). New methods are used to locate the spectrum of the linearized operator \(Lu=u''-cu'+f'(u^\ast)u\), where \(u^\ast\) is a traveling-wave solution with critical speed. In particular, \(L\) is studied on weighted spaces which are relatively smaller than \(L^\infty\) or \(L^p\). The main result applies, for instance, to equations with the well-known Fisher-KPP-type nonlinearity.
scientific article; zbMATH DE number 5141171