Existence results of steady-states of semilinear reaction-diffusion equations and their applications
This paper is concerned with the asymptotic behavior of solutions to the initial boundary value problem of semilinear parabolic equation \[ u_ t=u_{xx}+f(x,u,u_ x),\quad in\quad (a,b)\times (0,\infty) \] \[ u(a,t)=h(t),u(b,t)=k(t)\quad for\quad t>0;\quad u(x,0)=g(x)\quad for\quad a\leq x\leq b. \] Using the well known Nagumo lemma, the authors prove the existence of a steady-state solution for the above problem. Some results on quenching phenomena and existence of the minimal and maximal solutions are also presented. Sufficient conditions on the initial data which give rise to the phenomena of finite-time blow-up and the decay of the solution are also given, respectively.
scientific article; zbMATH DE number 4122471