This paper is concerned with semilinear reaction-diffusion systems under nonlinear dynamical boundary conditions. We will prove that such problems are well-posed on some Bessel potential spaces and that the corresponding solution defines a local semiflow on these spaces. This result will enable us to investigate the dynamical properties of the solutions under discussion. In particular, we will prove some conclusions concerning global existence and blow up phenomena as well as singular perturbation results.

In this paper we analyze nonlinear systems whose vector fields are driven by both an ordinary control $v$ and the derivative $\dot u$ of a second control $u$. It is clear that the presence of the derivative $\dot u$ causes some problems in the attempt to define a robust notion of trajectory corresponding to a discontinuous control $u$. On one hand, the nonlinearity of the systems under consideration makes a mere extension of the notion of solution by means of a measure--theoretical approach impossible. On the other hand, the interaction occurring between the derivative $\dot u$ of the {\it impulsive} control $u$ and the ordinary control $v$ prevent us from applying the results of [5], where a space--time extension of the concept of solution was already considered. Indeed in [5] the control $(u,v)$ had equibounded variation, whereas in the present work $v$ is just a measurable bounded map. After giving a stable notion of (space--time) solution to our problem we investigate the topological structure of the set of trajectories: as a consequence of having suppressed the restraint on the variation of $v$ we find that this set is no longer compact. This motivates the search for conditions under which the set of trajectories turns out to be closed again. Finally, in view of several applications we also analyze the case where the derivative $\dot u$ is constrained to range over a closed cone.

The Daleckij-Krein method for constructing a quadratic Lyapunov function for the equation $f'=Df(t)$ in Hilbert space is extended to include the case of an unbounded operator $D$ that generates a $C_0$-group. The extension is applied to obtain a quadratic Lyapunov function for the case of a group of weighted composition operators generated by a flow on a compact metric space together with a cocycle over this flow. These results are used to characterize the hyperbolicity of linear skew-product flows in terms of the existence of such a Lyapunov function. Also, the "trajectorial" method for constructing the Lyapunov function is discussed. Interrelations with Schrödinger, Riccati and Hamiltonian equations are discussed and an application to geodesic flows on two-dimensional Riemannian manifolds is given.

We consider the gap equation in superconductivity theory with a positive kernel. Under reasonable conditions on the kernel function, we prove that there exists at most one positive solution. This together with the known existence result gives a fairly complete understanding of the gap equation.

Our purpose is to construct a space of distribution solutions in a simple case for which the data is not smooth. We consider the mixed boundary value problem for the equation $-\Delta u=f$ in a domain $\Omega$ with polygonal boundary $\Gamma$. On each side of $\Gamma$ we impose either Dirichlet or Neumann boundary conditions. This is a ``corner problem'' whose solution contains corner singularities that are well understood [2, 3]. We are therefore in a position to construct a dual theory of distribution solutions for this mixed problem. In this paper we make this construction for distribution solutions $u \in L_2(\Omega)$; that is, the case when the solution is one step below the energy space in regularity. For this, we must give a careful definition of the data space associated with the mixed problem, and the "trace space" associated with a function $u \in L_2(\Omega)$. We find that there is always a distribution solution to the mixed boundary value problem, but the solution may not be unique; there may be distribution solutions to the homogeneous problem constructed with the help of the corner singular functions.

We consider the nonlinear Volterra integrodifferential equation $$ u_t - a* \text{div}\, h(\,\text{grad}\, u)=a*g, $$ where $x\in \mathbb{R}$, $t\geq 0$ and where the initial function $u(0,x)=w(x)$ is given. The kernel $a$ satisfies $a\in L_{\text{loc}} ^1(\mathbb{R}^{+})$ and the parabolicity condition $\mathfrak{R} \tilde a(\omega )\geq q | \Im \tilde a (\omega) | $ for some $q > 0$ and $\omega \in \mathbb{R}$. We suppose that for some $p> 4$ $$ g\in Y= L^2(\mathbb{R};L^2(\mathbb{R}^n)) \cap L^{p,\infty}(\mathbb{R}; H^{n-1}(\mathbb{R}^n)), $$ where $L^{p,\infty}(\mathbb{R};X)\overset{\rm {def}} \to = \{f : \sup_{T\geq 0}\int ^{T+1}_T \|f\|^p_X \! < \! \infty \}.$ It is shown that for $\|g\|_Y+\|\text{grad\,} w \|_{H^n(\mathbb{R}^n)}$ sufficiently small there exists a solution $u$, defined on $\mathbb{R} \times \mathbb{R}$ and satisfying $u_t,\Delta u\in Y$.

This paper is concerned with the fixed point index of a compact operator and its application to the study of multiple steady-state solutions of nonlinear reaction-diffusion systems. The method is simplified under the condition that the Banach space $X$ can be decomposed as $X=Y\oplus S_\varphi$, which is frequently satisfied by various reaction-diffusion models. A new method for proving the existence of positive steady-state solutions is developed by using this simplified method to semiflows. The result is applied to a three-species ecological model for which some sufficient conditions for the existence of positive steady-state solutions are obtained.

We prove that positive solutions of $-\Delta u = \lambda f(u)$ in $\Omega$ and $u = 0$ on $\partial \Omega$ where $f$ is increasing, concave, and $f(0) < 0$ satisfy $c_{1} \leq {\lambda f(d) \over d} \leq c_{2}$ where $d = \sup u.$ Also, we show that solutions of the above have exactly one inflection point.

We will study the existence of solutions of $$ -(\alpha^{p-1}(t)|u'|^{p-2}u')'= f(t,u)+h(t) \quad \text{in $[0,1]$}, $$ subject to various boundary conditions and $p>1$. In addition, we will give a detailed characterization of the eigenvalues and the Fucik spectrum of the corresponding differential operators. We use the Sturm comparison theorem and degree theory.

We obtain, via a Galerkin argument, the existence of a unique weak solution to the initial-boundary value problem for an incompressible bipolar viscous fluid satisfying nonhomogeneous boundary conditions. The analysis depends on the derivation of several key a priori estimates. Regularity results are also established and the solution is proven to be asymptotically stable when the forcing function and initial and boundary data decay in an appropriate sense.