Exact rates of convergence as t → +∞ for solutions of nonlinear evolution equations

Abstract.

We study the asymptotic behavior of solutions for a class of evolution problems in a Hilbert space, including those of the form

$$ u'(t) + \partial \varphi (u(t)) \mathrel\backepsilon 0,\;{\text{a}}{\text{.e}}{\text{. for }}t \in (0,\infty ), $$

where ∂φ is the subdifferential of a lower-semicontinuous convex function φ. We prove that if φ is coercive and locally sub-homogeneous of degree p, p ≥ 2, with respect to all the points of the set F of its minimizers, then solutions converge as t → +∞ to their limit in F at rates which are exactly of exponential type if p = 2, and of algebraic type if p > 2. Applications to nonlinear PDEs, including the two-phase Stefan problem in a bounded domain in R ⁿ, are given.