We consider a class of reaction-diffusion systems with macroscopic convection and nonlinear diffusion plus a nonstandard boundary condition which results as a model for heterogeneous catalysis in a stirred multiphase chemical reactor. Since the appearance of $T$-periodic feeds is a common feature in such applications, we study the problem of existence of a $T$-periodic solution. The model under consideration admits an abstract formulation in an appropriate $L^1$-setting, which leads to an evolution problem of the type \[ u' + Au \ni f(t,u) \ \mbox{ on } \ \mathbb R_+. \] Here $A$ is an $m$-accretive operator in a Banach space $X$ and $f:\mathbb R_+ \times K \to X$ is $T$-periodic and of Carathéodory type where $K$ is a closed, bounded, convex subset of $X$. Sufficient conditions on $A$, $f$ and $K$ to assure existence of $T$-periodic mild solutions for this evolution problem are provided and applied to the class of reaction-diffusion systems mentioned above.

We provide an elementary approach to integrable systems associated with hyperelliptic curves of infinite genus. In particular, we explore the extent to which the classical Burchnall-Chaundy theory generalizes in the infinite genus limit, and systematically study the effect of Darboux transformations for the KdV hierarchy on such infinite-genus curves. Our approach applies to complex-valued periodic solutions of the KdV hierarchy and naturally identifies the Riemann surface familiar from standard Floquet theoretic considerations with a limit of Burchnall-Chaundy curves.

We consider the following nonlinear degenerate parabolic equation which arises in some recent problems of mathematical finance: $$ \partial_{xx} u + u \partial_{y} u - \partial_{t} u =f. $$ Using a harmonic analysis technique on Lie groups, we prove that, if the solution $u$ satisfies condition $\partial_x u \neq 0$ in an open set $\Omega \subset \mathbb R^3$ and $f \in C^{\infty}(\Omega)$, then $u \in C^{\infty}(\Omega)$.

In this paper we complete the theory of metasolutions already developed in [8], [7], and [13] by showing how those metasolutions can be approached by means of classical solutions; metasolutions are large solutions extended to infinity. This result has important consequences not only from the point of view of the abstract theory, but also from the point of view of the numerical computation of the metasolutions in real applications.

In this paper we characterize the existence and prove the uniqueness of the stable positive steady-state for a general class of superlinear indefinite reaction diffusion equations in the absence of $L_\infty$ a priori bounds. More precisely, it will be shown that the model possesses a linearly stable positive steady-state if, and only if, the trivial solution is linearly unstable and the model possesses some positive steady-state. Moreover, it is unique if it exists. Actually, the minimal positive steady-state provides us with the unique linearly stable positive steady-state of the model. This is an extremely striking result since these problems can have an arbitrarily large number of positive steady-states as a result of having spatial inhomogeneities or varying the geometry of the support domain where the reaction takes place.