Superconvexity of the evolution operator and parabolic eigenvalue problems on ${\bf R}^n$
Daniel Daners, Pablo Koch‐Medina · Differential and Integral Equations · 1994
The purpose of this paper is to investigate the stability of the zero solution of the equationas the parameter ).. varies over JR+.Here we assume that the diffusion coefficient k: lR --t lR is a smooth and strictly positive T-periodic function (T > 0 a fixed number) and the weight function m:JRN x lR --t lR is smooth and T-periodic in the second argument (for the precise smoothness conditions consult Section 6).Furthermore, we shall assume that m changes sign and that m(x, t) ::::; -c 0. We remark that by suitable rescaling of time we could assume without loss of generality that k = 1.Stability shall be understood as stability with respect to the L00-norm and initial values in C0 (JRN), the space of continuous functions vanishing at infinity.More precisely, we shall interpret (1.1) as an abstract evolution equation in the Banach space Xo := (Co(lRN), ll•lloo)• This is accomplished by setting X1 := D(A) := {u E Xo: D.