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### Open Access Journals

DCDS-S

We consider operators in divergence form, $A_1u=(au')'$, and in nondivergence form, $A_2u=au''$, provided that the coefficient $a$ vanishes in an interior point of the space domain. Characterizing the domain of the operators, we prove that, under suitable assumptions, the operators $A_1$ and $A_2$, equipped with general Wentzell boundary conditions, are nonpositive and selfadjoint on spaces of $L^2$ type.

DCDS-S

Our aim in this paper is to define proper dynamic boundary conditions for a generalization
of the Cahn-Hilliard system proposed by M. Gurtin. Such boundary conditions take into account
the interactions with the walls in confined systems. We then study the existence and uniqueness of
weak solutions.

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 $C^{\infty}$.

DCDS

Jerome Arthur Goldstein (Jerry) was born on August 5, 1941 in Pittsburgh, PA. He attended Carnegie Mellon University, then called Carnegie Institute of Technology, where he earned his Bachelors of Science degree (1963), Masters of Science degree (1964) and Ph.D. (1967) in Mathematics. Jerry took a postdoctoral position at the Institute for Advanced Study in Princeton, followed by an Assistant Professorship at Tulane University. He became a Full Professor at Tulane University in 1975. After twenty-four years there, Jerry moved ``upriver" to join the faculty (and his wife, Gisèle) at Louisiana State University, where he was Professor of Mathematics from 1991 to 1996. In 1996 Jerry and Gisèle moved to the University of Memphis.

For more information please click the “Full Text” above.

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keywords:

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

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

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-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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