The semilinear Cauchy problem (1) u’(t) = Au(t) + G(u(t)), $u\left(0\right)=x\in \overline{D\left(A\right)}$, with a Hille-Yosida operator A and a nonlinear operator G: D(A) → X is considered under the assumption that ||G(x) - G(y)|| ≤ ||B(x -y )|| ∀x,y ∈ D(A) with some linear B: D(A) → X, $B{(\lambda -A)}^{-1}x=\lambda {\int }_0^{\infty }{e}^{-\lambda t}V\left(s\right)xds$, where V is of suitable small strong variation on some interval [0,ε). We will prove the existence of a semiflow on $[0,\infty )\times \overline{D\left(A\right)}$ that provides Friedrichs solutions in L₁ for (1). If X is a Banach lattice, we replace the condition above by |G(x) - G(y)| ≤ Bv whenever...

Soit $ℰ$ un espace topologique, ${ℰ}^{\text{'}}$ un espace métrique et $\left(S\right)$ un système d’équations d’évolution admettant une solution dans ${ℰ}^{\text{'}}$ pour toute donnée initiale dans $ℰ$ et stable vis-à-vis des données initiales sur $ℰ$. On montre que l’ensemble des données initiales pour lesquelles $\left(S\right)$ admet une unique solution est un $G_{\delta }$ de $ℰ$. En particulier, si l’unicité est vraie sur un sous-ensemble dense de $ℰ$, elle l’est génériquement.

In this paper we consider a general class of systems determined by operator valued measures which are assumed to be countably additive in the strong operator topology. This replaces our previous assumption of countable additivity in the uniform operator topology by the weaker assumption. Under the relaxed assumption plus an additional assumption requiring the existence of a dominating measure, we prove some results on existence of solutions and their regularity properties both for linear and semilinear...

Microstructures in phase-transitions of alloys are modeled by the energy minimization of a nonconvex energy density $\phi$. Their time-evolution leads to a nonlinear wave equation $u_{tt}=divS\left(Du\right)$ with the non-monotone stress-strain relation $S=D\phi$ plus proper boundary and initial conditions. This hyperbolic-elliptic initial-boundary value problem of changing types allows, in general, solely Young-measure solutions. This paper introduces a fully-numerical time-space discretization of this equation in a corresponding very...

Microstructures in phase-transitions of alloys are modeled by the energy minimization of a nonconvex energy density ϕ. Their time-evolution leads to a nonlinear wave equation $u_{tt}=\text{div}S\left(Du\right)$ with the non-monotone stress-strain relation $S=D\phi$ plus proper boundary and initial conditions. This hyperbolic-elliptic initial-boundary value problem of changing types allows, in general, solely Young-measure solutions. This paper introduces a fully-numerical time-space discretization of this equation in a corresponding...