Vanishing phenomena in fast decreasing generalized bistable equations

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Consider the reaction diffusion equation ut=uxx+f(u) with generalized bistable nonlinearity: f(0)=f(θ)=f(1)=0 for some θ∈(0,1), f(u)≤0 in (0,θ), f(u)>0 in (θ,1) and f(u)<0 in (1,∞). We show that when f(u) decreases sufficiently fast for u≫1, there exists ε0>0 such that, for any nonnegative initial data u0(x) with supp(u0)⊂[−ε0,ε0] (no matter how large ‖u0‖L∞ is), the solution u(x,t) to the Cauchy problem with initial data u0(x) always vanishes, that is, u(x,t)→0 as t→∞ in the L∞(R) norm.