The Carleman inequality for linear parabolic equations (see [5]) is extended to $L^q, 1\leq q\leq 2$ norm. Two applications pertaining maximum principle for the Bolza problem and stabilization of the semilinear heat equation are given.

This work is concerned with a diffusive model for some austenite-martensite phase transition processes ruled by the Cattaneo-Maxwell heat flux law, namely assuming for the heat flux a relaxed version of the classical Fourier law. A rigorous asymptotic analysis of the macroscopic model is performed and it is shown that such model is nothing but a singular perturbation of the standard Frémond model for shape memory alloys. Convergence results are proved along with error estimates.

Given an evolution family $\mathcal U:={(U(t,s))_{t\geq s}}$ on a Banach space $X$, we present some conditions under which asymptotic properties of $\mathcal U$ are stable under small perturbations by a family $$ \mathcal{B}:=(B(t),D(B(t))_{t\in\mathbb{J}},$$ $\mathbb{J} =\mathbb{R}$ or $\mathbb{R}_+$, of linear closed operators on $X$. Our results concern asymptotic properties like periodicity, (asymptotic) almost periodicity (even in the sense of Eberlein), uniform ergodicity and total uniform ergodicity. We present, moreover, an application of the abstract results to non-autonomous partial differential equations with delay.

In this paper we characterize the set of positive solutions of the nonlinear boundary value problem $v''(x)-\varepsilon v^{-q}(x)=0,$ $ 0 <x <1,$ $ v'(0)=0,$ $ v'(1)=v^p(1)$.

In this paper we perform the asymptotic analysis of a class of monotone quasilinear Neumann problems, with exponent $p\in ]1,+\infty[$ and nonstandard transmission condition, originating by change of variables from the quasilinear Neumann problem in a thin multidomain. This completes the $\Gamma$-convergence approach previously considered by the same authors. In particular, the corrector type result which is given here is more general.