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### Open Access Journals

DCDS-S

Let us consider the operator
$A_n u$:=$(-1)^{n+1} \alpha (x) u^{(2n)}$ on $H^n_0(0,1)$ with
domain $D(A_n)$:=$\{u\in H^n_0(0,1)\cap H^{2n}$

_{loc}$(0,1)\ :\ A_n u\in H^n_0(0,1)\}$, where $n\in\N$, $\alpha\in H^n_0(0,1)$, $\alpha (x)>0$ in $(0,1).$ Under additional boundedness and integrability conditions on $\alpha$ with respect to $x^{2n} (1-x)^{2n},$ we prove that $(A_n,D(A_n))$ is nonpositive and selfadjoint, thus it generates a cosine function, hence an analytic semigroup in the right half plane on $H^n_0(0,1)$. Analyticity results are also proved in $H^n (0,1).$ In particular, all results work well when $\alpha (x)=x^{j} (1-x)^{j}$ for $|j-n|<1/2$. Hardy type inequalities are also obtained.
CPAA

We study nonsymmetric second order elliptic operators with Wentzell boundary conditions in general domains with sufficiently smooth boundary. The ambient space is a space of $L^p$- type, $1\le p\le \infty$. We prove the existence of analytic quasicontractive $(C_0)$-semigroups generated by the closures of such operators, for any $1< p< \infty$. Moreover, we extend a previous result concerning the continuous dependence of these semigroups on the coefficients of the boundary condition.
We also specify precisely the domains of the generators explicitly in the case of bounded domains and $1 < p < \infty$, when all the ingredients of the problem, including the boundary of the domain, the coefficients, and the initial condition, are of class $C^{\infty}$.

CPAA

Let $\Omega$ be a smooth bounded domain in $R^N$ and let
\begin{eqnarray}
Lu=\sum_{j,k=1}^N \partial_{x_j}\left(a_{jk}(x)\partial_{x_k} u\right),
\end{eqnarray}
in $\Omega$ and
\begin{eqnarray}
Lu+\beta(x)\sum\limits_{j,k=1}^N a_{jk}(x)\partial_{x_j} u n_k+\gamma (x)u-q\beta(x)\sum_{j,k=1}^{N-1}\partial_{\tau_k}\left(b_{jk}(x)\partial_{\tau_j}u\right)=0,
\end{eqnarray}
on $\partial\Omega$
define a generalized Laplacian on $\Omega$ with a Wentzell boundary condition involving a generalized Laplace-Beltrami operator on the boundary.
Under some smoothness and positivity conditions on the coefficients, this defines a nonpositive selfadjoint operator, $-S^2$, on a suitable Hilbert space.
If we have a sequence of such operators $S_0,S_1,S_2,...$ with corresponding coefficients
\begin{eqnarray}
\Phi_n=(a_{jk}^{(n)},b_{jk}^{(n)}, \beta_n,\gamma_n,q_n)
\end{eqnarray}
satisfying $\Phi_n\to\Phi_0$ uniformly as $n\to\infty$, then $u_n(t)\to u_0(t)$ where $u_n$ satisfies
\begin{eqnarray}
i\frac{du_n}{dt}=S_n^m u_n,
\end{eqnarray}
or
\begin{eqnarray}
\frac{d^2u_n}{dt^2}+S_n^{2m} u_n=0,
\end{eqnarray}
or
\begin{eqnarray}
\frac{d^2u_n}{dt^2}+F(S_n)\frac{du_n}{dt}+S_n^{2m} u_n=0,
\end{eqnarray}
for $m=1,2,$ initial conditions independent of $n$, and for certain nonnegative functions $F$.
This includes Schrödinger equations, damped and undamped wave equations, and telegraph equations.

DCDS

This special issue consists of invited and carefully refereed papers
on specific topics related to evolution equations, semigroup theory
and related problems. Indeed, we thought that it would be very
valuable to produce such a volume on important and active areas of
research.

keywords:
XXXX

DCDS-S

Consider wave equations of the form

$\begin{align*}u''(t)+ A^2u(t)=0\end{align*}$ |

with $A$ an injective selfadjoint operator on a complex Hilbert space

. The kinetic, potential, and total energies of a solution $u$ are

$\mathcal{H}$ |

$\begin{align*}K(t)= \| u'(t)\|^2, P(t)= \|Au(t)\|^2, E(t) = K(t)+P(t).\end{align*}$ |

Finite energy solutions are those mild solutions for which

is finite. For such solutions

, that is, energy is conserved, and asymptotic equipartition of energy

$E(t)$ |

$E(t)= E(0)$ |

$\begin{align*}\lim_{t \longrightarrow ± ∞}K(t) = \lim_{t \longrightarrow ± ∞}P(t) = \frac{E(0)}{2}\end{align*}$ |

holds for all finite energy mild solutions iff

in the weak operator topology. In this paper we present the first extension of this result to the case where

is time dependent.

$e^{itA}\longrightarrow 0$ |

$A$ |

DCDS-S

In this paper we present a nonexistence result of exponentially
bounded positive solutions to a parabolic equation of Kolmogorov
type with a more general drift term perturbed by an inverse square
potential. This result generalizes the one obtained in [8].
Next we introduce some classes of nonlinear operators,
related to the filtration operators and the $p$-Laplacian, and involving Kolmogorov
operators. We establish the maximal monotonicity of some of these operators. In the third
part we discuss the possibility of some nonexistence results in the context of singular
potential perturbations of these nonlinear operators.

DCDS-B

We consider the realization of the operator $L_{\theta, a}u(x) $:$=
x^{2 a}u''(x) \ + \ (a x^{2 a - 1} + \theta x^a)u'(x)$, acting on
$C[0,+\infty]$, for $\theta\in\R$, $a\in\R$. We show that
$L_{\theta, a}$, with the so called Wentzell boundary conditions,
generates a Feller semigroup for any $\theta\in\R$, $a\in\R$. The
problem of finding optimal estimators for the corresponding
diffusion processes is also discussed, in connection with some
models in financial mathematics. Here $C[0,+\infty]$ is the space of
all real valued continuous functions on $[0,+\infty)$ which admit
finite limit at $+\infty$.

DCDS

Consider the Thomas-Fermi energy functional $E$ for a spin polarized atom or
molecule with $N_{1} $ [resp. $N_{2}$] spin up [resp. spin down]
electrons and total positive molecular charge Z. Incorporating the
Fermi-Amaldi correction as Benilan, Goldstein and Goldstein did, $E$ is not
convex. By replacing $E$ by a well-motivated convex minorant

**$ \mathcal{E}$**,we prove that**$ \mathcal{E} $**has a unique minimizing density $( \rho _{1},\rho _{2}) \ $ when $N_{1}+N_{2}\leq Z+1\ $and $N_{2}\ $is close to $N_{1}.$
DCDS

We find a simple sufficient criterion on a pair of nonnegative weight functions

and

on a Carnot group

so that the general weighted

Hardy type inequality

$V(x)$ |

$W(x) $ |

$\mathbb{G},$ |

$L^{p}$ |

$\begin{equation*}\int_{\mathbb{G}}V\left( x\right) \left\vert \nabla _{\mathbb{G}}\phi \left(x\right) \right\vert ^{p}dx\geq \int_{\mathbb{G}}W\left( x\right) \left\vert\phi \left( x\right) \right\vert ^{p}dx\end{equation*}$ |

is valid for any

and

It is worth noting here that our unifying method may be readily used both to recover most of the previously known weighted Hardy and Heisenberg-Pauli-Weyl type inequalities as well as to construct other new inequalities with an explicit best constant on

We also present some new results on two-weight

Hardy type inequalities with remainder terms on a bounded domain

in

via a differential inequality.

$φ ∈ C_{0}^{∞ }(\mathbb{G})$ |

$p>1.$ |

$\mathbb{G}.$ |

$L^{p}$ |

$Ω$ |

$\mathbb{G}$ |

DCDS-S

In the dynamic or Wentzell boundary condition for elliptic, parabolic and
hyperbolic partial differential equations, the positive flux coefficient $%
\beta $ determines the weighted surface measure $dS/\beta $ on the boundary
of the given spatial domain, in the appropriate Hilbert space that makes the
generator for the problem selfadjoint. Usually, $\beta $ is continuous and
bounded away from both zero and infinity, and thus $L^{2}\left( \partial
\Omega ,dS\right) $ and $L^{2}\left( \partial \Omega ,dS/\beta \right) $ are
equal as sets. In this paper this restriction is eliminated, so that both
zero and infinity are allowed to be limiting values for $\beta $. An
application includes the parabolic asymptotics for the Wentzell telegraph
equation and strongly damped Wentzell wave equation with general $\beta $.

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