We obtain an anti\-maximum principle for the following quasilinear parabolic problem: \begin{equation*} \tag*{\rm (P)} \left\{ \begin{alignedat}{2} \frac{\partial u}{\partial t} - \Delta_p u & = \lambda\, |u|^{p-2} u + f(x,t), & & \quad (x,t)\in \Omega\times (0,T); \\ u(x,t)& = 0, & & \quad (x,t)\in \partial\Omega\times (0,T); \\ u(x,0)& = u_0(x), & & \quad x\in \Omega, \end{alignedat} \right. \end{equation*} which involves the $p$-Laplace operator $\Delta_p u\equiv \mathrm{div}(|\nabla u|^{p-2}\nabla u)$ (with Dirichlet boundary conditions, $1 < p < \infty$) and a spectral parameter $\lambda\in \mathbb{R}^N$ taking values near the first eigenvalue $\lambda_1$ of $-\Delta_p$. We show that {\it any\/} weak solution $u\colon \Omega\times [0,T)\to \mathbb{R}$ of problem (P) (suitably defined in a standard way) eventually becomes positive for all $x\in \Omega$ and all times $t\geq T_{+}$, provided, for instance, $f(x,t)\geq \underline{f}(x) >0$ for some function $\underline{f}\in L^\infty(\Omega)$, $u_0\in W_0^{1,p}(\Omega)\cap L^{\infty}(\Omega)$, and $\lambda_1 < \lambda < \lambda_1 + \delta$. Here, the ``key'' constants $\delta\equiv \delta(\underline{f}, u_0) >0$ and $T_{+}\equiv T_{+}(f,u_0)\in (0,T)$ depend on $f$ (or $\underline{f}$ only) and $u_0$. In particular, a solution $u$ eventually becomes positive even if the initial data $u_0$ are "arbitrarily" negative as long as they are smooth enough.

This paper is devoted to the analysis of the asymptotic behaviour when the parameter $\lambda$ goes to $+\infty$ for operators of the form $-\Delta + \lambda a $ or, more generally, cooperative systems operators of the form $\left(\begin{smallmatrix} -\Delta+ {\lambda} a & -b \\ -c & -\Delta + {\lambda} d \end{smallmatrix} \right) $ where the potentials $a$ and $d$ vanish in some subregions of the domain $\Omega$. We use the theory of $\Gamma$-convergence, even for the non-variational cooperative system, to prove that for any reasonable bounded potentials $a$ and $d$ those operators converge in the strong resolvent sense to the operator in the vanishing regions of the potentials, as does the spectrum. The class of potentials considered here is fairly large, substantially improving previous results, allowing in particular ones that vanish on a Cantor set, and forcing us to enlarge the class of domains to the so-called quasi-open sets. For the system various situations are considered applying our general result to the interplay of the vanishing regions of the potentials of both equations.

In this paper we study a classical Dirichlet optimal control problem for a nonlinear elliptic equation with the coefficients as controls in $L^\infty(\Omega)$. Since such problems have no solutions in general, we make an assumption on the coefficients of the state equation and introduce the class of so-called solenoidal controls. Using the direct method in the calculus of variations, we prove the existence of at least one optimal pair. We also study the stability of the above optimal control problem with respect to the domain perturbation. With this goal we introduce the concept of Mosco-stability for such problems and analyze the variational properties of Mosco-stable problems with respect to different types of domain perturbations.

The aim of this paper is to establish time decay properties and dispersive estimates for strictly hyperbolic equations with homogeneous symbols and with time-dependent coefficients whose derivatives belong to $L^1(\mathbb{R})$. For this purpose, the method of asymptotic integration is developed for such equations and representation formulae for solutions are obtained. These formulae are analysed further to obtain time decay of $L^p$--$L^q$ norms of propagators for the corresponding Cauchy problems. It turns out that the decay rates can be expressed in terms of certain geometric indices of the limiting equation and we carry out a thorough analysis of this relation. This provides a comprehensive view of asymptotic properties of solutions to time-perturbations of hyperbolic equations with constant coefficients. Moreover, we also obtain the time decay rate of the $L^p$--$L^q$ estimates for equations of these kinds, so that the time well posedness of the corresponding nonlinear equations with additional semilinearity can be treated by standard Strichartz estimates.

In this paper a new notion of generalized solution to the initial-boundary-value problem for a strongly degenerate parabolic equation of the form $u_{t}+\nabla \cdot A(x,t,u)+B(x,t,u)={\varDelta} \beta(u)$ is treated. This type of solution is called a $BV$-entropy solution. Since equations of this form are linear combinations of time-dependent conservation laws and porous medium type equations, it is interesting to investigate interactions between singularities of solutions associated with the two different kinds of nonlinearities. However either of the part of conservation laws and that of porous medium type diffusion term may not be treated as a perturbation of the other. This observation leads us to the new notion of $BV$-entropy solution. Our objective here is to establish unique existence of such $BV$-entropy solutions under the homogeneous Neumann boundary conditions.