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We show that nonnegative solutions of $$\begin{array}{cc}& u_t-u_{xx}+f\left(u\right)=0,x\in ℝ,t>0,\hfill \\ & u=\alpha \overline{u},x\in ℝ,t=0,supp\overline{u}\text{compact}\hfill \end{array}$$ either converge to zero, blow up in $L^2$-norm, or converge to the ground state when $t\to \infty$, where the latter case is a threshold phenomenon when $\alpha >0$ varies. The proof is based on the fact that any bounded trajectory converges to a stationary solution. The function $f$ is typically nonlinear but has a sublinear growth at infinity. We also show that for superlinear $f$ it can happen that solutions converge to zero for any $\alpha >0$, provided $supp\overline{u}$ is sufficiently small.

We investigate the weak spectral mapping property (WSMP) $\overline{\mu ̂\left(\sigma \right(A\left)\right)}=\sigma \left(\mu ̂\left(A\right)\right)$, where A is the generator of a ₀-semigroup in a Banach space X, μ is a measure, and μ̂(A) is defined by the Phillips functional calculus. We consider the special case when X is a Banach algebra and the operators $e^{At}$, t ≥ 0, are multipliers.

Local well-posedness of the curve shortening flow, that is, local existence, uniqueness and smooth dependence of solutions on initial data, is proved by applying the Local Inverse Function Theorem and $L^2$-maximal regularity results for linear parabolic equations. The application of the Local Inverse Function Theorem leads to a particularly short proof which gives in addition the space-time regularity of the solutions. The method may be applied to general nonlinear evolution equations, but is presented...