We study a family of semilinear reaction-diffusion equations on spatial domains ${\Omega }_{\epsilon }$, ε > 0, in ${ℝ}^l$ lying close to a k-dimensional submanifold ℳ of ${ℝ}^l$. As ε → 0⁺, the domains collapse onto (a subset of) ℳ. As proved in [15], the above family has a limit equation, which is an abstract semilinear parabolic equation defined on a certain limit phase space denoted by $H{¹}_s\left(\Omega \right)$. The definition of $H{¹}_s\left(\Omega \right)$, given in the above paper, is very abstract. One of the objectives of this paper is to give more manageable characterizations...

In the paper, the problem of source function reconstruction in a differential equation of the parabolic type is investigated. Using the semigroup representation of trajectories of dynamical systems, we build a finite-step iterative procedure for solving this problem. The algorithm originates from the theory of closed-loop control (the method of extremal shift). At every step of the algorithm, the sum of a quality criterion and a linear penalty term is minimized. This procedure is robust to perturbations...

Using interpolation techniques we prove an optimal regularity theorem for the convolution $u\left(t\right)={\int }_0^{t}T\left(t-s\right)f\left(s\right)ds$, where $T\left(t\right)$ is a strongly continuous semigroup in general Banach space. In the case of abstract parabolic problems – that is, when $T\left(t\right)$ is an analytic semigroup – it lets us recover in a unified way previous regularity results. It may be applied also to some non analytic semigroups, such as the realization of the Ornstein-Uhlenbeck semigroup in $L^p\left({\mathbb{R}}^n\right)$, $1<p<\mathrm{\infty }$, in which case it yields new optimal regularity results in fractional...

We investigate the approximation of the evolution of compact hypersurfaces of ${ℝ}^N$ depending, not only on terms of curvature of the surface, but also on non local terms such as the measure of the set enclosed by the surface.

We investigate the approximation of the evolution of compact hypersurfaces of ${ℝ}^N$ depending, not only on terms of curvature of the surface, but also on non local terms such as the measure of the set enclosed by the surface.

The author studies the convective Cahn-Hilliard equation. Some results on the existence of classical solutions and asymptotic behavior of solutions are established. The instability of the traveling waves is also discussed.

In this paper we consider optimal control problems for abstract nonlinear evolution equations associated with time-dependent subdifferentials in a real Hilbert space. We prove the existence of an optimal control that minimizes the nonlinear cost functional. Also, we study approximating control problems of our equations. Then, we show the relationship between the original optimal control problem and the approximating ones. Moreover, we give some applications of our abstract results.

In this paper we consider a class of distributed parameter systems (partial differential equations) determined by strongly nonlinear operator valued measures in the setting of the Gelfand triple V ↪ H ↪ V* with continuous and dense embeddings where H is a separable Hilbert space and V is a reflexive Banach space with dual V*. The system is given by dx + A(dt,x) = f(t,x)γ(dt) + B(t)u(dt), x(0) = ξ, t ∈ I ≡ [0,T] where A is a strongly nonlinear operator valued measure...

We derive optimal regularity, in both time and space, for solutions of the Cauchy problem related to a degenerate differential equation in a Banach space X. Our results exhibit a sort of prevalence for space regularity, in the sense that the higher is the order of regularity with respect to space, the lower is the corresponding order of regularity with respect to time.