Stability theorem for singularly perturbed solutions to systems of reaction-diffusion equations
The author presents a stability theorem to singularly perturbed stationary solutions (SPS) of the systems of nonlinear diffusion equations with a small parameter \(\epsilon >0:\) \[ u_ t=\epsilon^ 2u_{xx}+f(u,v)\quad and\quad v_ t=Dv_{xx}+g(u,v),\quad x\in I=(0,1),\quad t>0,\quad u_ x=0=v_ x,\quad x\in \partial I=\{0,1\}. \] A new idea to solve the stability problem of SPS of one mode type for a general D is given. The singular limit eigenvalue problem: \[ \epsilon^ 2w_{xx}+f_ u^{\epsilon} w+f_ v^{\epsilon} z=\lambda w\quad and\quad Dz_{xx}+g_ u^{\epsilon} w+g_ v^{\epsilon} z=z,\quad in\quad I,\quad w_ x=0=z_ x\quad on\quad \partial I \] plays a key role. Authors' main conclusion is: Under suitable assumptions, there exists only one critical eigenvalue \(\lambda =\lambda_ 0(\epsilon)\) which is real and simple. When \(\epsilon\downarrow 0\), it behaves like \(\lambda_ 0(\epsilon)\simeq -\nu \epsilon\) \((\nu >0)\).
singularly perturbed stationary solutions
scientific article; zbMATH DE number 4017437
scientific article