American Institute of Mathematical Sciences

We are devoted with fractional abstract Cauchy problems. Required conditions on spaces and operators are given guaranteeing existence and uniqueness of solutions. An inverse problem is also studied. Applications from partial differential equations are given to illustrate the abstract fractional degenerate differential problems.

Here we present fractional univariate Ostrowski-Sugeno Fuzzy type inequalities. These are of Ostrowski-like inequalities in the setting of Sugeno fuzzy integral and its special-particular properties. In a fractional environment, they give tight upper bounds to the deviation of a function from its Sugeno-fuzzy averages. The fractional derivatives we use are of Canavati and Caputo types. This work is greatly inspired by [8], [1] and [2].

In this paper we prove the existence of global classical solutions to continuous coagulation–fragmentation equations with unbounded coefficients under the sole assumption that the coagulation rate is dominated by a power of the fragmentation rate, thus improving upon a number of recent results by not requiring any polynomial growth bound for either rate. This is achieved by proving a new result on the analyticity of the fragmentation semigroup and then using its regularizing properties to prove the local and then, under a stronger assumption, the global classical solvability of the coagulation–fragmentation equation considered as a semilinear perturbation of the linear fragmentation equation. Furthermore, we show that weak solutions of the coagulation–fragmentation equation, obtained by the weak compactness method, coincide with the classical local in time solutions provided the latter exist.

The aim of this paper is investigating the existence of at least one weak bounded solution of the quasilinear elliptic problem

\begin{document}$ \left\{ \begin{array}{ll} - {\rm{div}} (a(x,u,\nabla u)) + A_t(x,u,\nabla u)\ = \ f(x,u) &\hbox{in $\Omega$,}\\ u\ = \ 0 & \hbox{on $\partial\Omega$,} \end{array} \right. $\end{document}

where \begin{document}$ \Omega \subset \mathbb R^N $\end{document} is an open bounded domain and \begin{document}$ A(x,t,\xi) $\end{document}, \begin{document}$ f(x,t) $\end{document} are given real functions, with \begin{document}$ A_t = \frac{\partial A}{\partial t} $\end{document}, \begin{document}$ a = \nabla_\xi A $\end{document}.

We prove that, even if \begin{document}$ A(x,t,\xi) $\end{document} makes the variational approach more difficult, the functional associated to such a problem is bounded from below and attains its infimum when the growth of the nonlinear term \begin{document}$ f(x,t) $\end{document} is "controlled" by \begin{document}$ A(x,t,\xi) $\end{document}. Moreover, stronger assumptions allow us to find the existence of at least one positive solution.

We use a suitable Minimum Principle based on a weak version of the Cerami–Palais–Smale condition.

We prove the existence of infinitely many radial solutions to a Kirchhoff type problem in a ball with a super-cubic nonlinearity. Our methods rely on bifurcation analysis and energy estimates.

We consider an Ostrovsky-Hunter type equation, which also includes the short pulse equation, or the Kozlov-Sazonov equation. We prove the well-posedness of the entropy solution for the non-homogeneous initial boundary value problem. The proof relies on deriving suitable a priori estimates together with an application of the compensated compactness method.

In this paper, we define \begin{document}$ E_ \alpha(t^ \alpha A) $\end{document}, where \begin{document}$ A $\end{document} is the generator of an uniformly bounded (\begin{document}$ C_0 $\end{document}) semigroup and \begin{document}$ E_ \alpha(z) $\end{document} the Mittag-Leffler function. Since the mapping \begin{document}$ t\mapsto E_ \alpha(t^ \alpha A) $\end{document} has not the semigroup property, we cannot use the Trotter formula for representing the Feynman operator calculus. Thus for the Hamiltonian \begin{document}$ H_ \alpha = -\frac{{\hbar_ \alpha2}}{{2m}}\Delta +V(x) $\end{document}, we express \begin{document}$ E_ \alpha(t^ \alpha H_ \alpha ) $\end{document} by subordination principle of the Feynman path integral and we retrieve the corresponding Green function.

We study positive solutions to a steady state reaction diffusion equation arising in population dynamics, namely,

\begin{document}$ \begin{equation*} \label{abs} \left\lbrace \begin{matrix}-\Delta u = \lambda u(1-u) ;\; x\in\Omega\\ \frac{\partial u}{\partial \eta}+\gamma\sqrt{\lambda}[(A-u)^2+\epsilon]u = 0; \; x\in\partial \Omega \end{matrix} \right. \end{equation*} $\end{document}

where \begin{document}$ \Omega $\end{document} is a bounded domain in \begin{document}$ \mathbb{R}^N $\end{document}; \begin{document}$ N > 1 $\end{document} with smooth boundary \begin{document}$ \partial \Omega $\end{document} or \begin{document}$ \Omega = (0,1) $\end{document}, \begin{document}$ \frac{\partial u}{\partial \eta} $\end{document} is the outward normal derivative of \begin{document}$ u $\end{document} on \begin{document}$ \partial \Omega $\end{document}, \begin{document}$ \lambda $\end{document} is a domain scaling parameter, \begin{document}$ \gamma $\end{document} is a measure of the exterior matrix (\begin{document}$ \Omega^c $\end{document}) hostility, and \begin{document}$ A\in (0,1) $\end{document} and \begin{document}$ \epsilon>0 $\end{document} are constants. The boundary condition here represents a case when the dispersal at the boundary is U-shaped. In particular, the dispersal is decreasing for \begin{document}$ u<A $\end{document} and increasing for \begin{document}$ u>A $\end{document}. We will establish non-existence, existence, multiplicity and uniqueness results. In particular, we will discuss the occurrence of an Allee effect for certain range of \begin{document}$ \lambda $\end{document}. When \begin{document}$ \Omega = (0,1) $\end{document} we will provide more detailed bifurcation diagrams for positive solutions and their evolution as the hostility parameter \begin{document}$ \gamma $\end{document} varies. Our results indicate that when \begin{document}$ \gamma $\end{document} is large there is no Allee effect for any \begin{document}$ \lambda $\end{document}. We employ a method of sub-supersolutions to obtain existence and multiplicity results when \begin{document}$ N>1 $\end{document}, and the quadrature method to study the case \begin{document}$ N = 1 $\end{document}.

We consider a differential operator of order 2\begin{document}$ n $\end{document} of the type \begin{document}$ A_n u = (-1)^n (a u^{(n)})^{(n)} $\end{document}, where \begin{document}$ a(x)>0 $\end{document} in \begin{document}$ [0, 1]\setminus\{x_0\} $\end{document} and \begin{document}$ a(x_0) = 0 $\end{document}. We show that, for any \begin{document}$ n\in{\mathbb{N}} $\end{document}, the operator \begin{document}$ -A_n $\end{document} generates a contractive analytic semigroup of angle \begin{document}$ \pi/2 $\end{document} on \begin{document}$ L^2 (0, 1) $\end{document}. Note that the domain of \begin{document}$ A_n $\end{document} depends on the type of degeneracy of \begin{document}$ a $\end{document}. Our theorems extend some previous results in [3] where \begin{document}$ n = 1 $\end{document}.

Using a unified approach employing a homogeneous Lippmann-Schwinger-type equation satisfied by resonance functions and basic facts on Riesz potentials, we discuss the absence of threshold resonances for Dirac and Schrödinger operators with sufficiently short-range interactions in general space dimensions.

More specifically, assuming a sufficient power law decay of potentials, we derive the absence of zero-energy resonances for massless Dirac operators in space dimensions \begin{document}$ n \geqslant 3 $\end{document}, the absence of resonances at \begin{document}$ \pm m $\end{document} for massive Dirac operators (with mass \begin{document}$ m > 0 $\end{document}) in dimensions \begin{document}$ n \geqslant 5 $\end{document}, and recall the well-known case of absence of zero-energy resonances for Schrödinger operators in dimension \begin{document}$ n \geqslant 5 $\end{document}.

We study mixed hyperbolic systems with dynamic and Wentzell boundary conditions. The boundary condition contains a tangential operator which is strongly elliptic on the boundary. We prove results of generation of strongly continuous groups and well-posedness.

The problem of determining equity volatility from a knowledge of American option prices for a range of exercise (strike) prices and expirations is solved by minimization of a convex functional.

We prove a cone-type criterion for a boundary point to be regular for the Dirichlet problem related to (possibly) degenerate Ornstein–Uhlenbeck operators in \begin{document}$ \mathbb{R}^N $\end{document}. Our result extends the classical Zaremba cone criterion for the Laplace operator.

Suppose that \begin{document}$ u(x) $\end{document} is a positive subsolution to an elliptic equation in a bounded domain \begin{document}$ D $\end{document}, with the \begin{document}$ C^2 $\end{document} smooth boundary \begin{document}$ \partial D $\end{document}. We prove a quantitative version of the Hopf maximum principle that can be formulated as follows: there exists a constant \begin{document}$ \gamma>0 $\end{document} such that \begin{document}$ \partial_{\bf n}u(\tilde x) $\end{document} – the outward normal derivative at the maximum point \begin{document}$ \tilde x\in \partial D $\end{document} (necessary located at \begin{document}$ \partial D $\end{document}, by the strong maximum principle) – satisfies \begin{document}$ \partial_{\bf n}u(\tilde x)>\gamma u(\tilde x) $\end{document}, provided the coefficient \begin{document}$ c(x) $\end{document} by the zero order term satisfies \begin{document}$ \sup_{x\in D}c(x) = -c_*<0 $\end{document}. The constant \begin{document}$ \gamma $\end{document} depends only on the geometry of \begin{document}$ D $\end{document}, uniform ellipticity bound, \begin{document}$ L^\infty $\end{document} bounds on the coefficients, and \begin{document}$ c_* $\end{document}. The key tool used is the Feynman–Kac representation of a subsolution to the elliptic equation.

We consider the surface diffusion and Willmore flows acting on a general class of (possibly non–compact) hypersurfaces parameterized over a uniformly regular reference manifold possessing a tubular neighborhood with uniform radius. The surface diffusion and Willmore flows each give rise to a fourth–order quasilinear parabolic equation with nonlinear terms satisfying a specific singular structure. We establish well–posedness of both flows for initial surfaces that are \begin{document}$ C^{1+\alpha} $\end{document}–regular and parameterized over a uniformly regular hypersurface. For the Willmore flow, we also show long–term existence for initial surfaces which are \begin{document}$ C^{1+\alpha} $\end{document}–close to a sphere, and we prove that these solutions become spherical as time goes to infinity.

In this paper we study local and global in time existence for a class of nonlinear evolution equations having order eventually greater than 2 and not integer. The linear operator has an homogeneous damping term; the nonlinearity is of polynomial type without derivatives:

\begin{document}$ u_{tt}+ (-\Delta)^{2\theta}u+2\mu(-\Delta)^\theta u_t + |u|^{p-1}u = 0, \quad t\geq0, \ x\in {\mathbb{R}}^n, $\end{document}

with \begin{document}$ \mu>0 $\end{document}, \begin{document}$ \theta>0 $\end{document}. Since we are treating an absorbing nonlinear term, large data solutions can be considered.

We develop a model for the spatial spread of epidemic outbreak in a geographical region. The goal is to understand how spatial heterogeneity influences the transmission dynamics of the susceptible and infected populations. The model consists of a system of partial differential equations, which indirectly describes the disease transmission caused by the disease pathogen. The model is compared to data for the seasonal influenza epidemics in Puerto Rico for 2015-2016.

This paper provides two different extensions of a previous joint work "Time asymptotics of structured populations with diffusion and dynamic boundary conditions; Discrete Cont Dyn Syst, Series B, 23 (10) (2018)" devoted to asynchronous exponential asymptotics for bounded and weakly compact reproduction operators. The first extension considers bounded non weakly compact reproduction operators while the second extension deals with unbounded kernel reproduction operators and needs, as a preliminary step, a new generation result.

The computational powers of Mathematica are used to prove polynomial identities that are essential to obtain growth estimates for subdiagonal rational Padé approximations of the exponential function and to obtain new estimates of the constants of the Brenner-Thomée Approximation Theorem of Semigroup Theory.