Existence of a mild solution to a semilinear Cauchy problem with an almost sectorial operator is studied. Under additional regularity assumptions on the nonlinearity and initial data we also prove the existence of a classical solution to this problem. An example of a parabolic problem in Hölder spaces illustrates the abstract result.

We derive in this article some models of Cahn-Hilliard equations in nonisotropic media. These models, based on constitutive equations introduced by Gurtin in [19], take the work of internal microforces and also the deformations of the material into account. We then study the existence and uniqueness of solutions and obtain the existence of finite dimensional attractors.

We study stability and integrability of linear non-autonomous evolutionary Cauchy-problem $$\left(\mathrm{P}\right)\left\{\begin{array}{c}\dot{u}\left(t\right)+A\left(t\right)u\left(t\right)=f\left(t\right)t\text{-a.e.}\text{on}[0,\tau ],u\left(0\right)=0,\hfill \end{array}\right.$$ where $A:[0,\tau ]\to ℒ(X,D)$ is a bounded and strongly measurable function and $X$, $D$ are Banach spaces such that $D\underset{d}{\to }\hookrightarrow X$. Our main concern is to characterize $L^p$-maximal regularity and to give an explicit approximation of the problem (P).

We survey recent results on the mathematical modeling of nonconvex and nonsmooth contact problems arising in mechanics and engineering. The approach to such problems is based on the notions of an operator subdifferential inclusion and a hemivariational inequality, and focuses on three aspects. First we report on results on the existence and uniqueness of solutions to subdifferential inclusions. Then we discuss two classes of quasi-static hemivariational ineqaulities, and finally, we present ideas...