Advances in Differential Equations
___________________________________________________________________________________________
Advances in Differential Equations
Volume 30, Numbers 5-6, May/June 2025
Compact embeddings for radial Sobolev spaces of two variable exponents and nonlinear elliptic problems with variable critical exponent
Takao Ohno, Megumi Sano, and Tetsu Shimomura
Advances in Differential Equations, Volume 30, Number 5/6 (2025), 237-268
Interplay of geometric constraint and bonding force in the emergent behaviors of relativistic Cucker-Smale flocks
Hyunjin Ahn, Junhyeok Byeon, Seung-Yeal Ha, and Jaeyoung Yoon
Advances in Differential Equations, Volume 30, Number 5/6 (2025), 269-314
An expression for the solution of the Weinstein-Type equation by Clifford analysis
Xin Guo and Heju Yang
Advances in Differential Equations, Volume 30, Number 5/6 (2025), 315-334
Beris-Edwards models on evolving surfaces: A Lagrange-D'Alembert approach
Ingo Nitschke and Axel Voigt
Advances in Differential Equations, Volume 30, Number 5/6 (2025), 335-420
To appear in the next issue
Volume 30, Numbers 7-8, July/August 2025
Bifurcation results for nonlinear eigenvalue problems
involving the $(p,q)$-Laplace operator
Emmanuel Wend-Benedo Zongo and Bernhard Ruf
Advances in Differential Equations, Volume 30, Number 7/8 (2025), 421-454
Abstract:
In this paper, we analyze an eigenvalue problem
for nonlinear elliptic operators involving homogeneous
Dirichlet boundary conditions in an open
smooth bounded domain.
We prove bifurcation results from trivial solutions
and from infinity for considered nonlinear
eigenvalue problem. We also show the existence of
multiple solutions of the nonlinear problem using
variational methods.
The defocusing nonlinear Schrödinger equation with step-like oscillatory initial data
Samuel Fromm, Jonatan Lenells, and Ronald Quirchmayr
Advances in Differential Equations, Volume 30, Number 7/8 (2025), 455-525
Abstract:
We study the Cauchy problem for the
defocusing nonlinear
Schrödinger (NLS) equation under the
assumption that the solution vanishes as
$x \to \infty$ and approaches an
oscillatory plane wave as
$x \to -\infty$. We first develop an
inverse scattering transform formalism for
solutions satisfying such step-like boundary
conditions. Using this formalism, we prove
that there exists a global solution of the
corresponding Cauchy problem and establish
a representation for this solution in terms
of the solution of a Riemann--Hilbert problem.
By performing a steepest descent analysis of
this Riemann--Hilbert problem, we identify
three asymptotic sectors in the half-plane
$t\ge0$ of the
$xt$-plane and derive asymptotic formulas
for the solution in each of these sectors.
Finally, by restricting the constructed
solutions to the half-line
$x \ge 0$, we find a class of solutions with
asymptotically time-periodic boundary
values previously sought for in the context
of the NLS half-line problem.
The well-posedness of the stochastic nonlinear
Schrödinger
equations in $H^2(R^d)$
Isamu Dôku, Shunya Hashimoto, and Shuji Machihara
Advances in Differential Equations, Volume 30, Number 7/8 (2025), 527-560
Abstract:
The Cauchy problem for the stochastic nonlinear
Schrödinger
equation with multiplicative noise
is considered where the nonlinear term is of
power type and the noise coefficients are purely
imaginary numbers.
The main purpose of this paper is to construct
classical solutions in
$H^2(\mathbb{R}^d)$ for the problem.
The techniques of Kato [21, 22]
work well in overcoming smoothness problems
even for the stochastic equations.
Multi-bump solutions for the double phase critical
Schrödinger
equations involving logarithmic nonlinearity
Lifeng Guo, Sihua Liang, Bohui Lin, and Patrizia Pucci
Advances in Differential Equations, Volume 30, Number 7/8 (2025), 561-600
Abstract:
In this paper,
we consider the existence and multiplicity of
multi-bump solutions for the
following double phase critical
Schrödinger
equations with
logarithmic nonlinearity in
$\mathbb{R}^N$
\begin{equation}\label{E}
\begin{cases}
-\Delta_pu-\Delta_qu+(\lambda\mathcal {V}(x)
+\mathcal {Z}(x))(|u|^{p-2}u+|u|^{q-2}u) \\
\qquad\qquad\qquad\qquad\qquad\qquad
=\alpha |u|^{p-2}u\log |u|^p+|u|^{p^\ast-2}u, \\
u\in W^{1,p}(\mathbb{R}^N)\bigcap W^{1,q}
(\mathbb{R}^N),
\end{cases}
\tag{$\mathcal E$}
\end{equation}
where
$N\geq3, \alpha\in (1, \infty), \Delta_\wp
u=$div$(|\nabla|^{\wp-2}\nabla u)$
is the
$\wp$-Laplacian operator
with
$\wp\in\{p,q\}$,
$0 < q < p < p^\ast,$
the parameter
$\lambda\geq1$
and
$
\mathcal {Z},\,
\mathcal {V}: \mathbb{R}^N \rightarrow
\mathbb{R}$
are nonnegative continuous functions. Applying
variational methods, we prove {the} existence of at least
$2^k-1$
multi-bump solutions of
\eqref{E}
as
$\lambda\geq1$
is sufficiently
large. The main features and novelty of the
paper are the presence of the double
phase operator and of the logarithmic nonlinearity.
Moreover, these results
are new even in the case
$p = q =2$.
In a sense, we fill in the
blanks of previous papers in the subject.
To appear in the next issue
Volume 30, Numbers 9-10, September/October 2025
Energy decay for semilinear evolution equations with memory and time-dependent time delay feedback
Elisa Continelli and Cristina Pignotti
Advances in Differential Equations, Volume 30, Number 9/10 (2025), 601-634
Abstract:
In this paper, we study well-posedness and
exponential stability for semilinear
second-order evolution equations with memory
and time-varying delay feedback.
The time delay function is assumed to be continuous
and bounded. Under a suitable assumption on
the delay feedback, we are able to prove that
solutions corresponding to small initial data
are globally defined and satisfy an
exponential decay estimate.
Singular elliptic equations having a gradient term with natural growth
Adele Ferone, Anna Mercaldo, and Sergio Segura de Leon
Advances in Differential Equations, Volume 30, Number 9/10 (2025), 635-670
Abstract:
We study a class of Dirichlet boundary
value problems whose prototype is
\begin{equation}\label{1.2abs}
\left\{\begin{array}{ll}
-\Delta_p u =h(u)|\nabla u|^p+u^{q-1}+f(x)\,
&\quad\hbox{in } \ \Omega\,,\\
u\ge 0\,,&{\quad\hbox{in } \ \Omega}\\
u = 0\,
&\quad\hbox{on }\partial \Omega\,,
\end{array}\right.
\end{equation}
where
$\Omega$ an open bounded subset of
$\mathbb{R}^N$,
$0 < q < 1$,
$1 < p < N$,
$h$ is a continuous function and $f$ belongs
to a suitable Lebesgue space.
The main features of this problem are
the presence of a singular term and a
first order term with natural
growth in the gradient.
A priori estimates and existence
results are proved depending on the
summability of the datum $f$.
On the Föppl-von Kármán theory for elastic prestrained films with varying thickness
Hui Li
Advances in Differential Equations, Volume 30, Number 9/10 (2025), 671-706
Abstract:
We derive the variational limiting
theory of thin films,
parallel to the Föppl-von Kármán
theory in the nonlinear elasticity,
for films that have been prestrained
and whose thickness
is a general non-constant function.
Using
$\Gamma$-convergence,
we extend the existing results to the
variable thickness setting,
calculate the associated Euler-Lagrange
equations of the limiting energy,
and analyze convergence of equilibria.
The resulting formulas display the
interrelation between
deformations of the geometric mid-surface
and components of the growth tensor.
Maximal regularity for degenerate multi-term fractional integro-differential equations and applications
Rafael Aparicio and Valentin Keyantuo
Advances in Differential Equations, Volume 30, Number 9/10 (2025), 707-758
Abstract:
Many problems in the applied sciences can be
formulated as multi-term differential or
integro-differential equations.
We use the theory of operator-valued
Fourier multipliers to obtain
characterizations for well-posedness of
degenerate multi-term fractional
differential equations with infinite delay
in Banach spaces. We adopt the right-sided
Liouville fractional derivative on the real
axis and the Lebesgue-Bochner spaces
$L^p(\mathbb{R},X), \, 1\le p < \infty$ where
$X$ is a Banach space.
When $X$ is a UMD (Banach space in
which martingale differences are unconditional)
and $1 < p < \infty$, concrete conditions are
given for well-posedness using the concept of
$\mathcal{R}$-boundedness. Applications are
given for concrete integro-differential
equations which appear in several models
in the applied sciences, particularly
physics, biological sciences, rheology,
and material science. We present several
concrete examples of equations that can be
treated by the results obtained.
The equations considered in the applications
can contain fractional derivatives
in the space variable.
Copyright © 2025
Khayyam Publishing, Inc.
