Advances in Differential Equations Articles (Project Euclid)
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The latest articles from Advances in Differential Equations on Project Euclid, a site for mathematics and statistics resources.en-usCopyright 2012 Cornell University LibraryEuclid-L@cornell.edu (Project Euclid Team)Fri, 14 Dec 2012 11:51 ESTFri, 14 Dec 2012 11:51 ESThttp://projecteuclid.org/collection/euclid/images/logo_linking_100.gifProject Euclid
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Harnack's inequality for parabolid De Girogi classes in metric spaces
http://projecteuclid.org/euclid.ade/1355502220
<strong>Juha Kinnunen</strong>, <strong>Niko Marola</strong>, <strong>Michele Miranda, Jr.</strong>, <strong>Fabio Paronetto</strong><p><strong>Source: </strong>Adv. Differential Equations, Volume 17, Number 9-10, 801--832.</p><p><strong>Abstract:</strong><br/>
In this paper we study problems related to parabolic partial differential
equations in metric measure spaces equipped with a doubling measure and
supporting a Poincaréinequality. We give a definition of parabolic De Giorgi
classes and compare this notion with that of parabolic quasiminimizers. The main
result, after proving the local boundedness, is a scale- and location-invariant
Harnack inequality for functions belonging to parabolic De Giorgi classes. In
particular, the results hold true for parabolic quasiminimizers.
</p>projecteuclid.org/euclid.ade/1355502220_Fri, 14 Dec 2012 11:51 ESTFri, 14 Dec 2012 11:51 ESTInsensitizing controls for the Boussinesq system with no control on the temperature equationhttp://projecteuclid.org/euclid.ade/1487386868<strong>N. Carreño</strong>. <p><strong>Source: </strong>Advances in Differential Equations, Volume 22, Number 3/4, 235--258.</p><p><strong>Abstract:</strong><br/>
In this paper, we prove the existence of controls insensitizing the
$L^2$-norm of the solution of the Boussinesq system. The novelty here
is that no control is used on the temperature equation. Furthermore,
the control acting on the fluid equation can be chosen to have one
vanishing component. It is well known that the insensitizing control
problem is equivalent to a null controllability result for a cascade
system, which is obtained thanks to a suitable Carleman estimate for
the adjoint of the linearized system and an inverse mapping theorem.
The particular form of the adjoint equation will allow us to obtain
the null controllability of the linearized system.
</p>projecteuclid.org/euclid.ade/1487386868_20170217220119Fri, 17 Feb 2017 22:01 ESTA well posedness result for generalized solutions of Hamilton-Jacobi equationshttp://projecteuclid.org/euclid.ade/1487386869<strong>Sandro Zagatti</strong>. <p><strong>Source: </strong>Advances in Differential Equations, Volume 22, Number 3/4, 258--304.</p><p><strong>Abstract:</strong><br/>
We study the Dirichlet problem for stationary
Hamilton-Jacobi equations
$$
\begin{cases}
H(x, u(x), \nabla u(x)) = 0 & \ \textrm{in} \ \Omega \\
u(x)=\varphi(x) & \ \textrm{on} \ \partial \Omega.
\end{cases}
$$
We consider a Caratheodory hamiltonian $H=H(x,u,p)$, with a
Sobolev-type
(but not continuous) regularity
with respect to the space variable $x$, and prove existence and
uniqueness
of a Lipschitz continuous maximal generalized solution which, in the
continuous
case, turns out to be the classical viscosity solution.
In addition, we prove a continuous dependence property of the solution
with respect to the boundary datum $\varphi$, completing
in such a way a well posedness theory.
</p>projecteuclid.org/euclid.ade/1487386869_20170217220119Fri, 17 Feb 2017 22:01 ESTOn the local pressure of the Navier-Stokes equations and related systemshttp://projecteuclid.org/euclid.ade/1489802453<strong>Jörg Wolf</strong>. <p><strong>Source: </strong>Advances in Differential Equations, Volume 22, Number 5/6, 305--338.</p><p><strong>Abstract:</strong><br/>
In the study of local regularity of weak solutions to systems
related to incompressible viscous fluids local energy estimates
serve as important ingredients. However, this requires certain
information on the pressure. This fact has been used by
V. Scheffer in the notion of a suitable weak solution
to the Navier-Stokes equations, and in the proof of
the partial regularity due to Caffarelli, Kohn and Nirenberg.
In general domains, or in case of complex viscous fluid
models a global pressure does not necessarily exist.
To overcome this problem, in the present paper we construct
a local pressure distribution by showing that every
distribution $ \partial _t \boldsymbol u +\boldsymbol F $, which vanishes on the
set of smooth solenoidal vector fields can be represented by
a distribution $ \partial _t \nabla p_h +\nabla p_0 $,
where $\nabla p_h \sim \boldsymbol u $ and $ \nabla p_0 \sim \boldsymbol F$.
</p>projecteuclid.org/euclid.ade/1489802453_20170317220105Fri, 17 Mar 2017 22:01 EDTClassical solutions of the generalized Camassa-Holm equationhttp://projecteuclid.org/euclid.ade/1489802454<strong>John Holmes</strong>, <strong>Ryan C. Thompson</strong>. <p><strong>Source: </strong>Advances in Differential Equations, Volume 22, Number 5/6, 339--362.</p><p><strong>Abstract:</strong><br/>
In this paper, well-posedness in $C^1(\mathbb R)$ (a.k.a. classical solutions)
for a generalized Camassa-Holm equation (g-$k$$b$CH) having
$(k+1)$-degree nonlinearities is shown. This result holds for the
Camassa-Holm, the Degasperis-Procesi and the Novikov equations,
which improves upon earlier results in Sobolev and Besov spaces.
</p>projecteuclid.org/euclid.ade/1489802454_20170317220105Fri, 17 Mar 2017 22:01 EDTThe Cauchy problem for the shallow water type equations in low regularity spaces on the circlehttp://projecteuclid.org/euclid.ade/1489802455<strong>Wei Yan</strong>, <strong>Yongsheng Li</strong>, <strong>Xiaoping Zhai</strong>, <strong>Yimin Zhang</strong>. <p><strong>Source: </strong>Advances in Differential Equations, Volume 22, Number 5/6, 363--402.</p><p><strong>Abstract:</strong><br/>
In this paper, we investigate the Cauchy problem
for the shallow water type equation
\begin{align*}
u_{t}+\partial_{x}^{3}u
+ \tfrac{1}{2}\partial_{x}(u^{2})+\partial_{x}
(1-\partial_{x}^{2})^{-1}\left[u^{2}+\tfrac{1}{2}
u_{x}^{2}\right]=0, \ \ x\in {\mathbf T}={\mathbf R}/2\pi
\lambda,
\end{align*}
with low regularity data and $\lambda\geq1$. By applying the bilinear
estimate in $W^{s}$, Himonas and Misiołek (Commun. Partial Diff. Eqns.,
23 (1998), 123-139) proved that the problem is locally
well-posed in $H^{s}([0, 2\pi))$ with $s\geq {1}/{2}$
for small initial data. In this paper, we show that, when $s < {1}/{2}$, the bilinear
estimate in $W^{s}$ is invalid. We also demonstrate that the bilinear
estimate in $Z^{s}$ is indeed valid for ${1}/{6} < s < {1}/{2}$.
This enables us to prove that the
problem is locally well-posed in $H^{s}(\mathbf{T})$ with
${1}/{6} < s < {1}/{2}$ for small initial data.
</p>projecteuclid.org/euclid.ade/1489802455_20170317220105Fri, 17 Mar 2017 22:01 EDTConstant sign Green's function for simply supported beam equationhttp://projecteuclid.org/euclid.ade/1489802456<strong>Alberto Cabada</strong>, <strong>Lorena Saavedra</strong>. <p><strong>Source: </strong>Advances in Differential Equations, Volume 22, Number 5/6, 403--432.</p><p><strong>Abstract:</strong><br/>
The aim of this paper consists on the study of the following fourth-order
operator:
\begin{equation*}
T[M]\,u(t)\equiv u^{(4)}(t)+p_1(t)u'''(t)+p_2(t)u''(t)+Mu(t),\ t\in I \equiv
[a,b] ,
\end{equation*}
coupled with the two point boundary conditions:
\begin{equation*}
u(a)=u(b)=u''(a)=u''(b)=0 .
\end{equation*}
So, we define the following space:
\begin{equation*}
X=\left\lbrace u\in C^4(I) : u(a)=u(b)=u''(a)=u''(b)=0 \right\rbrace .
\end{equation*}
Here, $p_1\in C^3(I)$ and $p_2\in C^2(I)$.
By assuming that the second order linear differential equation
\begin{equation*}
L_2\, u(t)\equiv u''(t)+p_1(t)\,u'(t)+p_2(t)\,u(t)=0\,,\quad t\in I,
\end{equation*}
is disconjugate on $I$, we characterize the parameter's set
where the Green's function related to operator $T[M]$ in $X$
is of constant sign on $I \times I$. Such a characterization
is equivalent to the strongly inverse positive (negative)
character of operator $T[M]$ on $X$ and comes from the
first eigenvalues of operator $T[0]$ on suitable spaces.
</p>projecteuclid.org/euclid.ade/1489802456_20170317220105Fri, 17 Mar 2017 22:01 EDTThreshold and strong threshold solutions of a semilinear parabolic equationhttp://projecteuclid.org/euclid.ade/1493863418<strong>Pavol Quittner</strong>. <p><strong>Source: </strong>Advances in Differential Equations, Volume 22, Number 7/8, 433--456.</p><p><strong>Abstract:</strong><br/>
If $p>1+2/n$, then the equation $u_t-\Delta u = u^p, \quad x\in{\mathbb R}^n,\ t>0,$
possesses both positive global solutions and positive solutions which blow up in finite
time. We study the large time behavior of radial positive solutions lying on the
borderline between global existence and blow-up.
</p>projecteuclid.org/euclid.ade/1493863418_20170503220352Wed, 03 May 2017 22:03 EDTThe Cauchy problem on large time for a Boussinesq-Peregrine equation with large
topography variationshttp://projecteuclid.org/euclid.ade/1493863419<strong>Mesognon-Gireau Benoit</strong>. <p><strong>Source: </strong>Advances in Differential Equations, Volume 22, Number 7/8, 457--504.</p><p><strong>Abstract:</strong><br/>
We prove, in this paper, a long time existence result for a modified Boussinesq-Peregrine
equation in dimension $1$, describing the motion of Water Waves in shallow water, in the
case of a non flat bottom. More precisely, the dimensionless equations depend strongly on
three parameters $\epsilon,\mu,\beta$ measuring the amplitude of the waves, the
shallowness and the amplitude of the bathymetric variations, respectively. For the
Boussinesq-Peregrine model, one has small amplitude variations ($\epsilon = O(\mu)$). We
first give a local existence result for the original Boussinesq Peregrine equation as
derived by Boussinesq ([9], [8]) and Peregrine ([22]) in all dimensions. We then introduce
a new model which has formally the same precision as the Boussinesq-Peregrine equation,
and give a local existence result in all dimensions. We finally prove a local existence
result on a time interval of size $\frac{1}{\epsilon}$ in dimension $1$ for this new
equation, without any assumption on the smallness of the bathymetry $\beta$, which is an
improvement of the long time existence result for the Boussinesq systems in the case of
flat bottom ($\beta=0$) by [24].
</p>projecteuclid.org/euclid.ade/1493863419_20170503220352Wed, 03 May 2017 22:03 EDTLocal Hardy and Rellich inequalities for sums of squares of vector fieldshttp://projecteuclid.org/euclid.ade/1493863420<strong>Michael Ruzhansky</strong>, <strong>Durvudkhan Suragan</strong>. <p><strong>Source: </strong>Advances in Differential Equations, Volume 22, Number 7/8, 505--540.</p><p><strong>Abstract:</strong><br/>
We prove local refined versions of Hardy's and Rellich's inequalities as well as of
uncertainty principles for sums of squares of vector fields on bounded sets of smooth
manifolds under certain assumptions on the vector fields. We also give some explicit
examples, in particular, for sums of squares of vector fields on Euclidean spaces and for
sub-Laplacians on stratified Lie groups.
</p>projecteuclid.org/euclid.ade/1493863420_20170503220352Wed, 03 May 2017 22:03 EDTStable and unstable manifolds for quasilinear parabolic problems with fully nonlinear
dynamical boundary conditionshttp://projecteuclid.org/euclid.ade/1493863421<strong>Roland Schnaubelt</strong>. <p><strong>Source: </strong>Advances in Differential Equations, Volume 22, Number 7/8, 541--592.</p><p><strong>Abstract:</strong><br/>
We develop a wellposedness and regularity theory for a large class of quasilinear
parabolic problems with fully nonlinear dynamical boundary conditions. Moreover, we
construct and investigate stable and unstable local invariant manifolds near a given
equilibrium. In a companion paper, we treat center, center-stable and center-unstable
manifolds for such problems and investigate their stability properties. This theory
applies e.g. to reaction-diffusion systems with dynamical boundary conditions and to the
two-phase Stefan problem with surface tension.
</p>projecteuclid.org/euclid.ade/1493863421_20170503220352Wed, 03 May 2017 22:03 EDTAnalyticity of semigroups generated by higher order elliptic operators in spaces of
bounded functions on $C^{1}$ domainshttp://projecteuclid.org/euclid.ade/1495850455<strong>Takuya Suzuki</strong>. <p><strong>Source: </strong>Advances in Differential Equations, Volume 22, Number 9/10, 593--620.</p><p><strong>Abstract:</strong><br/>
This paper shows the analyticity of semigroups generated by higher order divergence type
elliptic operators in $L^{\infty}$ spaces in $C^{1}$ domains which may be unbounded. For
this purpose, we establish resolvent estimates in $L^{\infty}$ spaces by a contradiction
argument based on a blow-up method. Our results yield the $L^{\infty}$ analyticity of
solutions of parabolic equations for $C^{1}$ domains.
</p>projecteuclid.org/euclid.ade/1495850455_20170526220106Fri, 26 May 2017 22:01 EDTPositive solutions of Schrödinger equations and Martin boundaries for skew product
elliptic operatorshttp://projecteuclid.org/euclid.ade/1495850456<strong>Minoru Murata</strong>, <strong>Tetsuo Tsuchida</strong>. <p><strong>Source: </strong>Advances in Differential Equations, Volume 22, Number 9/10, 621--692.</p><p><strong>Abstract:</strong><br/>
We consider positive solutions of elliptic partial differential equations on non-compact
domains of Riemannian manifolds. We establish general theorems which determine Martin
compactifications and Martin kernels for a wide class of elliptic equations in skew
product form, by thoroughly exploiting parabolic Martin kernels for associated parabolic
equations developed in [35] and [25]. As their applications, we explicitly determine the
structure of all positive solutions to a Schrödinger equation and the Martin boundary
of the product of Riemannian manifolds. For their sharpness, we show that the Martin
compactification of ${\mathbb R}^2$ for some Schrödinger equation is so much
distorted near infinity that no product structures remain.
</p>projecteuclid.org/euclid.ade/1495850456_20170526220106Fri, 26 May 2017 22:01 EDTLocal stabilization of compressible Navier-Stokes equations in one dimension around
non-zero velocityhttp://projecteuclid.org/euclid.ade/1495850457<strong>Debanjana Mitra</strong>, <strong>Mythily Ramaswamy</strong>, <strong>Jean-Pierre Raymond</strong>. <p><strong>Source: </strong>Advances in Differential Equations, Volume 22, Number 9/10, 693--736.</p><p><strong>Abstract:</strong><br/>
In this paper, we study the local stabilization of one dimensional compressible
Navier-Stokes equations around a constant steady solution $(\rho_s, u_s)$, where
$\rho_s>0, u_s\neq 0$. In the case of periodic boundary conditions , we determine a
distributed control acting only in the velocity equation, able to stabilize the system,
locally around $(\rho_s, u_s)$, with an arbitrary exponential decay rate. In the case
of Dirichlet boundary conditions , we determine boundary controls for the velocity and for
the density at the inflow boundary, able to stabilize the system, locally around $(\rho_s,
u_s)$, with an arbitrary exponential decay rate.
</p>projecteuclid.org/euclid.ade/1495850457_20170526220106Fri, 26 May 2017 22:01 EDTOn local $L_p$-$L_q$ well-posedness of incompressible two-phase flows with phase
transitions: Non-equal densities with large initial datahttp://projecteuclid.org/euclid.ade/1495850458<strong>Senjo Shimizu</strong>, <strong>Shintaro Yagi</strong>. <p><strong>Source: </strong>Advances in Differential Equations, Volume 22, Number 9/10, 737--764.</p><p><strong>Abstract:</strong><br/>
A basic model for incompressible two-phase flows with phase transitions where the
interface is nearly flat in the case of non-equal densities is considered. Local
well-posedness of the model in $L_p$ in a time setting $L_q$ in a space setting was proved
in [30] under a smallness assumption for the initial data. In this paper, we remove the
smallness assumption for the initial data.
</p>projecteuclid.org/euclid.ade/1495850458_20170526220106Fri, 26 May 2017 22:01 EDTConvergence of the Allen-Cahn equation with constraint to Brakke's mean curvature
flowhttp://projecteuclid.org/euclid.ade/1495850459<strong>Keisuke Takasao</strong>. <p><strong>Source: </strong>Advances in Differential Equations, Volume 22, Number 9/10, 765--792.</p><p><strong>Abstract:</strong><br/>
In this paper, we consider the Allen-Cahn equation with constraint. In 1994, Chen and
Elliott [9] studied the asymptotic behavior of the solution of the Allen-Cahn equation
with constraint. They proved that the zero level set of the solution converges to the
classical solution of the mean curvature flow under the suitable conditions on initial
data. In 1993, Ilmanen [20] proved the existence of the mean curvature flow via the
Allen-Cahn equation without constraint in the sense of Brakke. We proved the same
conclusion for the Allen-Cahn equation with constraint.
</p>projecteuclid.org/euclid.ade/1495850459_20170526220106Fri, 26 May 2017 22:01 EDTThe variable coefficient thin obstacle problem: Higher regularityhttps://projecteuclid.org/euclid.ade/1504231224<strong>Herbert Koch</strong>, <strong>Angkana Rüland</strong>, <strong>Wenhui Shi</strong>. <p><strong>Source: </strong>Advances in Differential Equations, Volume 22, Number 11/12, 793--866.</p><p><strong>Abstract:</strong><br/>
In this article, we continue our investigation of the variable
coefficients thin obstacle problem which was initiated in
[20], [21]. Using a partial
Hodograph-Legendre transform and the implicit function
theorem, we prove the higher order Hölder regularity for the
regular free boundary, if the associated coefficients
are of the corresponding regularity. For the zero
obstacle, this yields an improvement of a full
derivative for the free boundary regularity compared to the
regularity of the coefficients. In the presence of inhomogeneities,
we gain three halves of a derivative for the free boundary
regularity with respect to the regularity of the inhomogeneity.
Further, we show analyticity of the regular free boundary for analytic
coefficients. We also discuss the set-up of $W^{1,p}$ coefficients
with $p>n+1$ and $L^p$ inhomogeneities.
Key ingredients in our analysis are the introduction of generalized
Hölder spaces, which allow to interpret the transformed fully
nonlinear, degenerate (sub)elliptic equation as a perturbation of the
Baouendi-Grushin operator, various uses of intrinsic geometries
associated with appropriate operators, the application of the
implicit function theorem to deduce (higher) regularity.
</p>projecteuclid.org/euclid.ade/1504231224_20170831220044Thu, 31 Aug 2017 22:00 EDTExistence, non-degeneracy of proportional positive solutions and least energy solutions for a fractional elliptic systemhttps://projecteuclid.org/euclid.ade/1504231225<strong>Qihan He</strong>, <strong>Shuangjie Peng</strong>, <strong>Yanfang Peng</strong>. <p><strong>Source: </strong>Advances in Differential Equations, Volume 22, Number 11/12, 867--892.</p><p><strong>Abstract:</strong><br/>
In this paper, we study the following fractional
nonlinear Schrödinger system
$$
\left\{
\begin{array}{ll}
(-\Delta)^s u +u=\mu_1 |u|^{2p-2}u+\beta |v|^p|u|^{p-2}u,
~~x\in \mathbb R^N, \\
(-\Delta)^s v +v=\mu_2 |v|^{2p-2}v+\beta |u|^p|v|^{p-2}v,
~~x\in \mathbb R^N,
\end{array}
\right.
$$
where
$0 < s < 1, \mu_1 > 0, \mu_2 > 0, 1 < p < 2_s^*/2, 2_s^*=+\infty$
for $N\le 2s$ and $2_s^*=2N/(N-2s)$ for $N > 2s$, and
$\beta \in \mathbb R$ is a coupling constant. We investigate
the existence and non-degeneracy of proportional
positive vector solutions for the above system
in some ranges of $\mu_1,\mu_2, p, \beta$.
We also prove that the least energy vector
solutions must be proportional and unique under some additional
assumptions.
</p>projecteuclid.org/euclid.ade/1504231225_20170831220044Thu, 31 Aug 2017 22:00 EDTExistence of solutions to a class of weakly coercive diffusion equations with singular initial datahttps://projecteuclid.org/euclid.ade/1504231226<strong>Marco Papi</strong>, <strong>Maria Michaela Porzio</strong>, <strong>Flavia Smarrazzo</strong>. <p><strong>Source: </strong>Advances in Differential Equations, Volume 22, Number 11/12, 893--962.</p><p><strong>Abstract:</strong><br/>
We prove existence of suitably defined measure-valued solutions to the homogeneous
Dirichlet initial-boundary value problem with a Radon measure
as initial datum, for a class of degenerate parabolic
equations without strong coerciveness.
The notion of solution is natural, since it is obtained by a
suitable approximation procedure which can be regarded
as a first step towards a continuous dependence on the
initial data. Moreover, we also discuss some qualitative
properties of the constructed solutions concerning the
evolution of their singular part.
</p>projecteuclid.org/euclid.ade/1504231226_20170831220044Thu, 31 Aug 2017 22:00 EDTConley index in Hilbert spaces versus the generalized topological degreehttps://projecteuclid.org/euclid.ade/1504231227<strong>Zbigniew Błaszczyk</strong>, <strong>Anna Gołębiewska</strong>, <strong>Sławomir Rybicki</strong>. <p><strong>Source: </strong>Advances in Differential Equations, Volume 22, Number 11/12, 963--982.</p><p><strong>Abstract:</strong><br/>
We prove a version of the Poincaré–Hopf theorem suitable
for strongly indefinite functionals and then apply it
to infer a number of bifurcation results in
infinite-dimensional Hilbert spaces.
</p>projecteuclid.org/euclid.ade/1504231227_20170831220044Thu, 31 Aug 2017 22:00 EDTMultiple solutions of a Kirchhoff type elliptic problem with the Trudinger-Moser growthhttps://projecteuclid.org/euclid.ade/1504231228<strong>D. Naimen</strong>, <strong>C. Tarsi</strong>. <p><strong>Source: </strong>Advances in Differential Equations, Volume 22, Number 11/12, 983--1012.</p><p><strong>Abstract:</strong><br/>
We consider a Kirchhoff type elliptic problem;
\begin{equation*}
\begin{cases}
-\left(1+\alpha \int_{\Omega}|\nabla u|^2dx\right)\Delta u
=f(x,u),\ u\ge0\text{ in }\Omega,\\
u=0\text{ on }\partial \Omega,
\end{cases}
\end{equation*}
where $\Omega\subset \mathbb{R}^2$ is a bounded domain with a
smooth boundary $\partial \Omega$, $\alpha > 0$ and $f$ is a
continuous function in $\overline{\Omega}\times \mathbb{R}$.
Moreover, we assume $f$ has the Trudinger-Moser growth. We prove
the existence of solutions of (P), so extending a former
result by de Figueiredo-Miyagaki-Ruf [11] for the case
$\alpha =0$ to the case $\alpha>0$. We emphasize that we also show
a new
multiplicity result induced by the nonlocal dependence. In order
to prove this, we carefully discuss the geometry of the associated
energy functional and the concentration compactness analysis for
the critical case.
</p>projecteuclid.org/euclid.ade/1504231228_20170831220044Thu, 31 Aug 2017 22:00 EDTTransport and equilibrium in non-conservative systemshttps://projecteuclid.org/euclid.ade/1508983358<strong>L. Chayes</strong>, <strong>H. K. Lei</strong>. <p><strong>Source: </strong>Advances in Differential Equations, Volume 23, Number 1/2, 1--64.</p><p><strong>Abstract:</strong><br/>
We study, in finite volume, a grand canonical version of the McKean-Vlasov equation where the total particle content is allowed to vary. The dynamics is anticipated to minimize an appropriate grand canonical free energy; we make this notion precise by introducing a metric on a set of positive Borel measures without pre-prescribed mass and demonstrating that the dynamics is a gradient flow with respect to this metric. Moreover, we develop a JKO-type scheme suitable for these problems. The latter ideas have general applicability to a class of second order non-conservative problems. For this particular system we prove, using the JKO-type scheme, that under certain conditions – not too far from optimal - convergence to the uniform stationary state is exponential with a rate which is independent of the volume. By contrast, in related conservative systems, decay rates scale (at best) with the square of the characteristic length of the system. This suggests that a grand canonical approach may be useful for both theoretical and computational study of large scale systems.
</p>projecteuclid.org/euclid.ade/1508983358_20171025220253Wed, 25 Oct 2017 22:02 EDTNonlinear travelling waves on complete Riemannian manifoldshttps://projecteuclid.org/euclid.ade/1508983360<strong>Mayukh Mukherjee</strong>. <p><strong>Source: </strong>Advances in Differential Equations, Volume 23, Number 1/2, 65--88.</p><p><strong>Abstract:</strong><br/>
We study travelling wave solutions to nonlinear Schrödinger and Klein-Gordon equations on complete Riemannian manifolds, which have a bounded Killing field $X$. For a natural class of power-type nonlinearities, we use standard variational techniques to demonstrate the existence of travelling waves on complete weakly homogeneous manifolds. If the manifolds in question are weakly isotropic, we prove that they have genuine subsonic travelling waves, at least for a non-empty set of parameters. Finally we establish that a slight perturbation of the Killing field $X$ will result in a controlled perturbation of the travelling wave solutions (in appropriate $L^p$-norms).
</p>projecteuclid.org/euclid.ade/1508983360_20171025220253Wed, 25 Oct 2017 22:02 EDTElliptic and parabolic equations with Dirichlet conditions at infinity on Riemannian manifoldshttps://projecteuclid.org/euclid.ade/1508983361<strong>P. Mastrolia</strong>, <strong>D. D. Monticelli</strong>, <strong>F. Punzo</strong>. <p><strong>Source: </strong>Advances in Differential Equations, Volume 23, Number 1/2, 89--108.</p><p><strong>Abstract:</strong><br/>
We investigate existence and uniqueness of bounded solutions of parabolic equations with unbounded coefficients in $M\times \mathbb R_+$, where $M$ is a complete noncompact Riemannian manifold. Under specific assumptions, we establish existence of solutions satisfying prescribed conditions at infinity, depending on the direction along which infinity is approached. We consider also elliptic equations on $M$ with similar conditions at infinity.
</p>projecteuclid.org/euclid.ade/1508983361_20171025220253Wed, 25 Oct 2017 22:02 EDTSign-changing solutions for non-local elliptic equations involving the fractional Laplacainhttps://projecteuclid.org/euclid.ade/1508983363<strong>Yinbin Deng</strong>, <strong>Wei Shuai</strong>. <p><strong>Source: </strong>Advances in Differential Equations, Volume 23, Number 1/2, 109--134.</p><p><strong>Abstract:</strong><br/>
In this paper, we consider the existence of sign-changing solutions for fractional elliptic equations of the form \begin{equation*} \left\{\begin{array}{ll} (-\Delta)^s u=f(x,u) & \text{in}\ \Omega , \\ u=0 & \text{in}\ \mathbb R^N\setminus \Omega, \end{array} \right. \end{equation*} where $s\in(0,1)$ and $\Omega\subset \mathbb R^N$ is a bounded smooth domain. Since the non-local operator $(-\Delta)^s$ is involved in the equation, the variational functional of the equation has totally different properties from the local cases. By introducing some new ideas, we prove, via variational method and the method of invariant sets of descending flow, that the problem has a positive solution, a negative solution and a sign-changing solution under suitable conditions. Moreover, if $f(x,u)$ satisfies a monotonicity condition, we show that the problem has a least energy sign-changing solution with its energy is strictly larger than that of the ground state solution of Nehari type. We also obtain an unbounded sequence of sign-changing solutions if $f(x,u)$ is odd in $u$.
</p>projecteuclid.org/euclid.ade/1508983363_20171025220253Wed, 25 Oct 2017 22:02 EDTAlmost global existence of weak solutions for the nonlinear elastodynamics system for a class of strain energieshttps://projecteuclid.org/euclid.ade/1508983364<strong>Sébastien Court</strong>, <strong>Karl Kunisch</strong>. <p><strong>Source: </strong>Advances in Differential Equations, Volume 23, Number 1/2, 135--160.</p><p><strong>Abstract:</strong><br/>
The aim of this paper is to prove the existence of almost global weak solutions for the unsteady nonlinear elastodynamics system in dimension $d=2$ or $3$, for a range of strain energy density functions satisfying some given assumptions. These assumptions are satisfied by the main strain energies generally considered. The domain is assumed to be bounded, and mixed boundary conditions are considered. Our approach is based on a nonlinear parabolic regularization technique, involving the $p$-Laplace operator. First we prove the existence of a local-in-time solution for the regularized system, by a fixed point technique. Next, using an energy estimate, we show that if the data are small enough, bounded by $\varepsilon >0$, then the maximal time of existence does not depend on the parabolic regularization parameter, and the behavior of the lifespan $T$ is $\gtrsim \log (1/\varepsilon)$, defining what we call here almost global existence . The solution is thus obtained by passing this parameter to zero. The key point of our proof is due to recent nonlinear Korn's inequalities proven by Ciarlet and Mardare in $W^{1,p}$ spaces, for $p>2$.
</p>projecteuclid.org/euclid.ade/1508983364_20171025220253Wed, 25 Oct 2017 22:02 EDTOn the focusing energy-critical fractional nonlinear Schrödinger equationshttps://projecteuclid.org/euclid.ade/1513652445<strong>Yonggeun Cho</strong>, <strong>Gyeongha Hwang</strong>, <strong>Tohru Ozawa</strong>. <p><strong>Source: </strong>Advances in Differential Equations, Volume 23, Number 3/4, 161--192.</p><p><strong>Abstract:</strong><br/>
We consider the fractional nonlinear Schrödinger equation (FNLS) with non-local dispersion $|\nabla|^{\alpha}$ and focusing energy-critical Hartree type nonlinearity $[-(|x|^{-2{\alpha}}*|u|^2)u]$. We first establish a global well-posedness of radial case in energy space by adopting Kenig-Merle arguments [20] when the initial energy and initial kinetic energy are less than those of ground state, respectively. We revisit and highlight long time perturbation, profile decomposition and localized virial inequality. As an application of the localized virial inequality, we provide a proof for finite time blowup for energy critical Hartree equations via commutator technique introduced in [2].
</p>projecteuclid.org/euclid.ade/1513652445_20171218220053Mon, 18 Dec 2017 22:00 ESTGlobal regularity of the 2D magnetic Bénard system with partial dissipationhttps://projecteuclid.org/euclid.ade/1513652446<strong>Zhuan Ye</strong>. <p><strong>Source: </strong>Advances in Differential Equations, Volume 23, Number 3/4, 193--238.</p><p><strong>Abstract:</strong><br/>
In this paper, we consider the Cauchy problem of the two-dimensional (2D) magnetic Bénard system with partial dissipation. On the one hand, we obtain the global regularity of the 2D magnetic Bénard system with zero thermal conductivity. The main difficulty is the zero thermal conductivity. To bypass this difficulty, we exploit the structure of the coupling system about the vorticity and the temperature and use the Maximal $L_t^{p}L_x^{q}$ regularity for the heat kernel. On the other hand, we also establish the global regularity of the 2D magnetic Bénard system with horizontal dissipation, horizontal magnetic diffusion and with either horizontal or vertical thermal diffusivity. This settles the global regularity issue unsolved in the previous works. Additionally, in the Appendix, we also show that with a full Laplacian for the diffusive term of the magnetic field and half of the full Laplacian for the temperature field, the global regularity result holds true as long as the power $\alpha$ of the fractional Laplacian dissipation for the velocity field is positive.
</p>projecteuclid.org/euclid.ade/1513652446_20171218220053Mon, 18 Dec 2017 22:00 ESTAsymptotics for the modified Boussinesq equation in one space dimensionhttps://projecteuclid.org/euclid.ade/1513652447<strong>Nakao Hayashi</strong>, <strong>Pavel I. Naumkin</strong>. <p><strong>Source: </strong>Advances in Differential Equations, Volume 23, Number 3/4, 239--294.</p><p><strong>Abstract:</strong><br/>
We consider the Cauchy problem for the modified Boussinesq equation in one space dimension \begin{equation*} \begin {cases} w_{tt}=a^{2}\partial _{x}^{2}w-\partial _{x}^{4}w+\partial _{x}^{2} ( w^{3} ) ,\text{ } ( t,x ) \in \mathbb{R}^{2}, \\ w ( 0,x ) =w_{0} ( x ) ,\text{ }w_{t} ( 0,x ) =w_{1} ( x ) ,\text{ }x\in \mathbb{R}\text{,} \end {cases} \end{equation*} where $a > 0.$ We study the large time asymptotics of solutions to the Cauchy problem for the modified Boussinesq equation. We apply the factorization technique developed recently in papers [5], [6], [7], [8].
</p>projecteuclid.org/euclid.ade/1513652447_20171218220053Mon, 18 Dec 2017 22:00 ESTThe Friedrichs extension for elliptic wedge operators of second orderhttps://projecteuclid.org/euclid.ade/1513652448<strong>Thomas Krainer</strong>, <strong>Gerardo A. Mendoza</strong>. <p><strong>Source: </strong>Advances in Differential Equations, Volume 23, Number 3/4, 295--328.</p><p><strong>Abstract:</strong><br/>
Let ${\mathcal M}$ be a smooth compact manifold whose boundary is the total space of a fibration ${\mathcal N}\to {\mathcal Y}$ with compact fibers, let $E\to{\mathcal M}$ be a vector bundle. Let \begin{equation} A:C_c^\infty( \overset{\,\,\circ} {\mathcal M};E)\subset x^{-\nu} L^2_b({\mathcal M};E)\to x^{-\nu} L^2_b({\mathcal M};E) $ \tag*{(†)} \end{equation} be a second order elliptic semibounded wedge operator. Under certain mild natural conditions on the indicial and normal families of $A$, the trace bundle of $A$ relative to $\nu$ splits as a direct sum ${\mathscr T}={\mathscr T}_F\oplus{\mathscr T}_{aF}$ and there is a natural map ${\mathfrak P} :C^\infty({\mathcal Y};{\mathscr T}_F)\to C^\infty( \overset{\,\,\circ} {\mathcal M};E)$ such that $C^\infty_{{\mathscr T}_F}({\mathcal M};E)={\mathfrak P} (C^\infty({\mathcal Y};{\mathscr T}_F)) +\dot C^\infty({\mathcal M};E)\subset {\mathcal D}_{\max}(A)$. It is shown that the closure of $A$ when given the domain $C^\infty_{{\mathscr T}_F}({\mathcal M};E)$ is the Friedrichs extension of (†) and that this extension is a Fredholm operator with compact resolvent. Also given are theorems pertaining the structure of the domain of the extension which completely characterize the regularity of its elements at the boundary.
</p>projecteuclid.org/euclid.ade/1513652448_20171218220053Mon, 18 Dec 2017 22:00 ESTRegularity and time behavior of the solutions of linear and quasilinear parabolic equationshttps://projecteuclid.org/euclid.ade/1516676481<strong>Maria Michaela Porzio</strong>. <p><strong>Source: </strong>Advances in Differential Equations, Volume 23, Number 5/6, 329--372.</p><p><strong>Abstract:</strong><br/>
In this paper, we study the regularity, the uniqueness and the asymptotic behavior of the solutions to a class of nonlinear operators in dependence of the summability properties of the datum $f$ and of the initial datum $u_0$. The case of only summable data $f$ and $u_0$ is allowed. We prove that these equations satisfy surprising regularization phenomena. Moreover, we prove estimates (depending continuously from the data) that for zero datum $f$ become well known decay (or ultracontractive) estimates.
</p>projecteuclid.org/euclid.ade/1516676481_20180122220130Mon, 22 Jan 2018 22:01 ESTRegularity and singularity of the blow-up curve for a wave equation with a derivative nonlinearityhttps://projecteuclid.org/euclid.ade/1516676482<strong>Takiko Sasaki</strong>. <p><strong>Source: </strong>Advances in Differential Equations, Volume 23, Number 5/6, 373--408.</p><p><strong>Abstract:</strong><br/>
We study a blow-up curve for the one dimensional wave equation $\partial_t^2 u- \partial_x^2 u = |\partial_t u|^p$ with $p>1$. The purpose of this paper is to show that the blow-up curve is a $C^1$ curve if the initial values are large and smooth enough. To prove the result, we convert the equation into a first order system, and then apply a modification of the method of Caffarelli and Friedman [2]. Moreover, we present some numerical investigations of the blow-up curves. From the numerical results, we were able to confirm that the blow-up curves are smooth if the initial values are large and smooth enough. Moreover, we can predict that the blow-up curves have singular points if the initial values are not large enough even they are smooth enough.
</p>projecteuclid.org/euclid.ade/1516676482_20180122220130Mon, 22 Jan 2018 22:01 ESTTraveling waves in a simplified gas-solid combustion model in porous mediahttps://projecteuclid.org/euclid.ade/1516676483<strong>Fatih Ozbag</strong>, <strong>Stephen Schecter</strong>, <strong>Grigori Chapiro</strong>. <p><strong>Source: </strong>Advances in Differential Equations, Volume 23, Number 5/6, 409--454.</p><p><strong>Abstract:</strong><br/>
We study the combustion waves that occur when air is injected into a porous medium containing initially some solid fuel and prove the existence of traveling waves using phase plane analysis. We also identify all the possible ways that combustion waves and contact discontinuities can combine to produce wave sequences that solve boundary value problems on infinite intervals with generic constant boundary data.
</p>projecteuclid.org/euclid.ade/1516676483_20180122220130Mon, 22 Jan 2018 22:01 ESTMountain pass solutions for the fractional Berestycki-Lions problemhttps://projecteuclid.org/euclid.ade/1516676484<strong>Vincenzo Ambrosio</strong>. <p><strong>Source: </strong>Advances in Differential Equations, Volume 23, Number 5/6, 455--488.</p><p><strong>Abstract:</strong><br/>
We investigate the existence of least energy solutions and infinitely many solutions for the following nonlinear fractional equation \begin{align*} (-\Delta)^{s} u = g(u) \mbox{ in } \mathbb R^{N}, \end{align*} where $s\in (0,1)$, $N\geq 2$, $(-\Delta)^{s}$ is the fractional Laplacian and $g: \mathbb R \rightarrow \mathbb R $ is an odd $\mathcal{C}^{1, \alpha}$ function satisfying Berestycki-Lions type assumptions. The proof is based on the symmetric mountain pass approach developed by Hirata, Ikoma and Tanaka in [33]. Moreover, by combining the mountain pass approach and an approximation argument, we also prove the existence of a positive radially symmetric solution for the above problem when $g$ satisfies suitable growth conditions which make our problem fall in the so called “zero mass” case.
</p>projecteuclid.org/euclid.ade/1516676484_20180122220130Mon, 22 Jan 2018 22:01 ESTLocal well-posedness and global existence for the biharmonic heat equation with exponential nonlinearityhttps://projecteuclid.org/euclid.ade/1526004064<strong>Mohamed Majdoub</strong>, <strong>Sarah Otsmane</strong>, <strong>Slim Tayachi</strong>. <p><strong>Source: </strong>Advances in Differential Equations, Volume 23, Number 7/8, 489--522.</p><p><strong>Abstract:</strong><br/>
In this paper, we prove local well-posedness in Orlicz spaces for the biharmonic heat equation $\partial_{t} u+ \Delta^2 u=f(u),\; t > 0,\; x \in \mathbb R^N,$ with $f(u)\sim \mbox{e}^{u^2}$ for large $u.$ Under smallness condition on the initial data and for exponential nonlinearity $f$ such that $|f(u)|\sim |u|^m$ as $u\to 0,$ $m\geq 2$, $N(m-1)/4\geq 2$, we show that the solution is global. Moreover, we obtain decay estimates for large time for the nonlinear biharmonic heat equation as well as for the nonlinear heat equation. Our results extend to the nonlinear polyharmonic heat equation.
</p>projecteuclid.org/euclid.ade/1526004064_20180510220112Thu, 10 May 2018 22:01 EDTWeak solvability for Dirichlet partial differential inclusions in Orlicz-Sobolev spaceshttps://projecteuclid.org/euclid.ade/1526004065<strong>Nicuşor Costea</strong>, <strong>Gheorghe Moroşanu</strong>, <strong>Csaba Varga</strong>. <p><strong>Source: </strong>Advances in Differential Equations, Volume 23, Number 7/8, 523--554.</p><p><strong>Abstract:</strong><br/>
We study PDI's of the type $-\Delta_\Phi u\in \partial_C f(x,u)$ subject to Dirichlet boundary condition in a bounded domain $\Omega\subset\mathbb{R}^N$ with Lipschitz boundary $\partial\Omega$. Here, $\Phi:\mathbb{R}\rightarrow [0,\infty)$ is the $N$-function defined by $\Phi(t):=\int_0^t a(|s|)s\,ds$, with $a:(0,\infty)\rightarrow (0,\infty)$ a prescribed function, not necessarily differentiable, and $\Delta_\Phi u:={\rm div}(a(|\nabla u|)\nabla u)$ is the $\Phi$-Laplacian. In addition, $f:\Omega\times\mathbb{R}\rightarrow \mathbb{R}$ is a locally Lipschitz function with respect to the second variable and $\partial_C$ denotes the Clarke subdifferential. Using a direct variational method and a nonsmooth version of the Mountain Pass Theorem the existence of nontrivial weak solutions is established. A multiplicity alternative is also proved without imposing an Ambrosetti-Rabinowitz type condition. More precisely, we show that our problem possesses either at least two nontrivial weak solutions or a rich family of negative eigenvalues. Several examples which highlight the applicability of our theoretical results are also provided.
</p>projecteuclid.org/euclid.ade/1526004065_20180510220112Thu, 10 May 2018 22:01 EDTExistence results for non-local elliptic systems with nonlinearities interacting with the spectrumhttps://projecteuclid.org/euclid.ade/1526004066<strong>Olímpio H. Miyagaki</strong>, <strong>Fábio Pereira</strong>. <p><strong>Source: </strong>Advances in Differential Equations, Volume 23, Number 7/8, 555--580.</p><p><strong>Abstract:</strong><br/>
In this work, we establish an existence result for a class of non-local variational elliptic systems with critical growth, but with nonlinearities interacting with the fractional laplacian spectrum. More specifically, we treat the situation when the interval defined by two eigenvalues of the real matrix coming from the linear part contains an eigenvalue of the spectrum of the fractional laplacian operator. In this case, there are situations where resonance or double resonance phenomena can occur. The novelty here is because, up to our knowledge, the results that have been appeared in the literature up to now, this interval does not intercept the fractional laplacian spectrum. The proof is made by using the linking theorem due to Rabinowitz.
</p>projecteuclid.org/euclid.ade/1526004066_20180510220112Thu, 10 May 2018 22:01 EDTDiffusion phenomena for the wave equation with space-dependent damping term growing at infinityhttps://projecteuclid.org/euclid.ade/1526004067<strong>Motohiro Sobajima</strong>, <strong>Yuta Wakasugi</strong>. <p><strong>Source: </strong>Advances in Differential Equations, Volume 23, Number 7/8, 581--614.</p><p><strong>Abstract:</strong><br/>
In this paper, we study the asymptotic behavior of solutions to the wave equation with damping depending on the space variable and growing at the spatial infinity. We prove that the solution is approximated by that of the corresponding heat equation as time tends to infinity. The proof is based on semigroup estimates for the corresponding heat equation having a degenerate diffusion at spatial infinity and weighted energy estimates for the damped wave equation. To construct a suitable weight function for the energy estimates, we study a certain elliptic problem.
</p>projecteuclid.org/euclid.ade/1526004067_20180510220112Thu, 10 May 2018 22:01 EDTGround and bound state solutions for a Schrödinger system with linear and nonlinear couplings in $\mathbb{R}^N$https://projecteuclid.org/euclid.ade/1526004068<strong>Kanishka Perera</strong>, <strong>Cyril Tintarev</strong>, <strong>Jun Wang</strong>, <strong>Zhitao Zhang</strong>. <p><strong>Source: </strong>Advances in Differential Equations, Volume 23, Number 7/8, 615--648.</p><p><strong>Abstract:</strong><br/>
We study the existence of ground and bound state solutions for a system of coupled Schrödinger equations with linear and nonlinear couplings in $\mathbb{R}^N$. By studying the limit system and using concentration compactness arguments, we prove the existence of ground and bound state solutions under suitable assumptions. Our results are new even for the limit system.
</p>projecteuclid.org/euclid.ade/1526004068_20180510220112Thu, 10 May 2018 22:01 EDTOn the modified scattering of $3$-d Hartree type fractional Schrödinger equations with Coulomb potential for any given initial and boundary data.https://projecteuclid.org/euclid.ade/1528855474<strong>Yonggeun Cho</strong>, <strong>Gyeongha Hwang</strong>, <strong>Changhun Yang</strong>. <p><strong>Source: </strong>Advances in Differential Equations, Volume 23, Number 9/10, 649--692.</p><p><strong>Abstract:</strong><br/>
In this paper, we study 3-d Hartree type fractional Schrödinger equations: $$ i\partial_{t}u-\vert\nabla\vert^{\alpha}u = \lambda \left ( |x|^{-\gamma} *\vert u\vert^{2} \right ) u,\;\;1 < \alpha < 2,\;\;0 < \gamma < 3,\; \lambda \in \mathbb R \setminus \{0\}. $$ In [7] it is known that no scattering occurs in $L^2$ for the long range ($0 < \gamma \le 1$). In [4, 10, 8] the short-range scattering ($1 < \gamma < 3$) was treated for the scattering in $H^s$. In this paper, we consider the critical case ($\gamma = 1$) and prove a modified scattering in $L^\infty$ on the frequency to the Cauchy problem with small initial data. For this purpose, we investigate the global behavior of $x e^{it |\nabla|^\alpha } u$, $x^2 e^{it |\nabla|^\alpha } u$ and $ \langle\xi\rangle ^5 \widehat{e^{it |\nabla|^\alpha } u}$. Due to the non-smoothness of $ |\nabla|^\alpha $ near zero frequency the range of $\alpha$ is restricted to $(\frac{17}{10}, 2)$.
</p>projecteuclid.org/euclid.ade/1528855474_20180612220444Tue, 12 Jun 2018 22:04 EDTGlobal existence for the heat flow of symphonic maps into sphereshttps://projecteuclid.org/euclid.ade/1528855476<strong>Masashi Misawa</strong>, <strong>Nobumitsu Nakauchi</strong>. <p><strong>Source: </strong>Advances in Differential Equations, Volume 23, Number 9/10, 693--724.</p><p><strong>Abstract:</strong><br/>
In our previous papers, we introduce symphonic maps ([9]) and show a Hölder continuity of symphonic maps from domains of $\mathbb{R}^4$ into the spheres ([6], [7]). In this paper, we consider the heat flow of symphonic maps with values into spheres and prove a global existence of a weak solution to the Cauchy-Dirichlet problem for any given initial and boundary data.
</p>projecteuclid.org/euclid.ade/1528855476_20180612220444Tue, 12 Jun 2018 22:04 EDTWell-posedness for the Cauchy problem of the Klein-Gordon-Zakharov system in five and higher dimensionshttps://projecteuclid.org/euclid.ade/1528855477<strong>Isao Kato</strong>, <strong>Shinya Kinoshita</strong>. <p><strong>Source: </strong>Advances in Differential Equations, Volume 23, Number 9/10, 725--750.</p><p><strong>Abstract:</strong><br/>
We study the Cauchy problem of the Klein-Gordon-Zakharov system in spatial dimension $d \ge 5$ with initial datum $(u, \partial_t u, n, \partial_t n)|_{t=0} \in H^{s+1}(\mathbb{R}^d) \times H^s(\mathbb{R}^d) \times \dot{H}^s(\mathbb{R}^d) \times \dot{H}^{s-1} (\mathbb{R}^d)$. The critical value of $s$ is $s_c=d/2-2$. By $U^2, V^2$ type spaces, we prove that the small data global well-posedness and scattering hold at $s=s_c$ in $d \ge 5$.
</p>projecteuclid.org/euclid.ade/1528855477_20180612220444Tue, 12 Jun 2018 22:04 EDTLong-time behavior of solutions to the fifth-order modified KdV-type equationhttps://projecteuclid.org/euclid.ade/1528855478<strong>Mamoru Okamoto</strong>. <p><strong>Source: </strong>Advances in Differential Equations, Volume 23, Number 9/10, 751--792.</p><p><strong>Abstract:</strong><br/>
We consider the long-time behavior of solutions to the fifth-order modified KdV-type equation. For the local-in-time well-posedness, we show that regularity conditions where by a tri-linear estimate holds in the Fourier restriction norm spaces, which is an extension of Kwon's result (2008). Using the method of testing by wave packets, we prove the small-data global existence and modified scattering. We derive the leading asymptotic in both the self-similar and oscillatory regions.
</p>projecteuclid.org/euclid.ade/1528855478_20180612220444Tue, 12 Jun 2018 22:04 EDTExtension theory approach in the stability of the standing waves for the NLS equation with point interactions on a star graphhttps://projecteuclid.org/euclid.ade/1537840834<strong>Jaime Angulo Pava</strong>, <strong>Nataliia Goloshchapova</strong>. <p><strong>Source: </strong>Advances in Differential Equations, Volume 23, Number 11/12, 793--846.</p><p><strong>Abstract:</strong><br/>
The aim of this work is to demonstrate the effectiveness of the extension theory of symmetric operators in the investigation of the stability of standing waves for the nonlinear Schrödinger equations with two types of non-linearities (power and logarithmic) and two types of point interactions ($\delta$- and $\delta'$-) on a star graph. Our approach allows us to overcome the use of variational techniques in the investigation of the Morse index for self-adjoint operators with non-standard boundary conditions which appear in the stability study. We also demonstrate how our method simplifies the proof of the stability results known for the NLS equation with point interactions on the line.
</p>projecteuclid.org/euclid.ade/1537840834_20180924220051Mon, 24 Sep 2018 22:00 EDTA classification for wave models with time-dependent potential and speed of propagationhttps://projecteuclid.org/euclid.ade/1537840835<strong>Marcelo Rempel Ebert</strong>, <strong>Wanderley Nunes do Nascimento</strong>. <p><strong>Source: </strong>Advances in Differential Equations, Volume 23, Number 11/12, 847--888.</p><p><strong>Abstract:</strong><br/>
In this paper, we study the long time behavior of energy solutions for a class of wave equation with time-dependent potential and speed of propagation. We introduce a classification of the potential term, which clarifies whether the solution behaves like the solution to the wave equation or Klein-Gordon equation. Moreover, the derived linear estimates are applied to obtain global (in time) small data energy solutions for the Cauchy problem to semilinear Klein-Gordon models with power nonlinearity.
</p>projecteuclid.org/euclid.ade/1537840835_20180924220051Mon, 24 Sep 2018 22:00 EDTOn a generalized fractional Du Bois-Reymond lemma and its applicationshttps://projecteuclid.org/euclid.ade/1537840836<strong>Rafał Kamocki</strong>. <p><strong>Source: </strong>Advances in Differential Equations, Volume 23, Number 11/12, 889--908.</p><p><strong>Abstract:</strong><br/>
In the paper, we derive a fractional version of the Du Bois-Reymond lemma for a generalized Riemann-Liouville derivative (derivative in the Hilfer sense). It is a generalization of well known results of such a type for the Riemann-Liouville and Caputo derivatives. Next, we use this lemma to investigate critical points of a some Lagrange functional (we derive the Euler-Lagrange equation for this functional).
</p>projecteuclid.org/euclid.ade/1537840836_20180924220051Mon, 24 Sep 2018 22:00 EDTSome $L^\infty$ solutions of the hyperbolic nonlinear Schrödinger equation and their stabilityhttps://projecteuclid.org/euclid.ade/1544497233<strong>Simão Correia</strong>, <strong>Mário Figueira</strong>. <p><strong>Source: </strong>Advances in Differential Equations, Volume 24, Number 1/2, 1--30.</p><p><strong>Abstract:</strong><br/>
Consider the hyperbolic nonlinear Schrödinger equation $\mathrm {(HNLS)}$ over $\mathbb R^d$ $$ iu_t + u_{xx} - \Delta_{\textbf{y}} u + \lambda |u|^\sigma u=0. $$ We deduce the conservation laws associated with $\mathrm {(HNLS)}$ and observe the lack of information given by the conserved quantities. We build several classes of particular solutions, including hyperbolically symmetric solutions , spatial plane waves and spatial plane waves , which never lie in $H^1$. Motivated by this, we build suitable functional spaces that include both $H^1$ solutions and these particular classes, and prove local well-posedness on these spaces. Moreover, we prove a stability result for both spatial plane waves and spatial standing waves with respect to small $H^1$ perturbations.
</p>projecteuclid.org/euclid.ade/1544497233_20181210220046Mon, 10 Dec 2018 22:00 ESTWell-posedness and large time behavior of solutions for the electron inertial Hall-MHD systemhttps://projecteuclid.org/euclid.ade/1544497234<strong>Yasuhide Fukumoto</strong>, <strong>Xiaopeng Zhao</strong>. <p><strong>Source: </strong>Advances in Differential Equations, Volume 24, Number 1/2, 31--68.</p><p><strong>Abstract:</strong><br/>
In this paper, the properties of weak and strong solutions for the Hall-magnetohydrodynamic system augmented by the effect of electron inertia are studied. First, we establish the existence and uniqueness of local-in-time strong solutions; Then, we prove the existence of global strong solutions under the condition that $\|u_0\|_{\dot{H}^{\frac12}}+\|B_0\|_{\dot{H}^{\frac12}} +\|\nabla B_0\|_{\dot{H}^{\frac12}}$ is sufficiently small. Moreover, by applying a cut-off function and generalized energy inequality, we show that the weak solution of electron inertia Hall-MHD system approaches zero as the time $t\rightarrow\infty$. Finally, the algebraic decay rate of the weak solution of electron inertia Hall-MHD system is established by using Fourier splitting method and the properties of decay character.
</p>projecteuclid.org/euclid.ade/1544497234_20181210220046Mon, 10 Dec 2018 22:00 ESTMultiscale Gevrey asymptotics in boundary layer expansions for some initial value problem with merging turning pointshttps://projecteuclid.org/euclid.ade/1544497235<strong>Alberto Lastra</strong>, <strong>Stéphane Malek</strong>. <p><strong>Source: </strong>Advances in Differential Equations, Volume 24, Number 1/2, 69--136.</p><p><strong>Abstract:</strong><br/>
We consider a nonlinear singularly perturbed PDE leaning on a complex perturbation parameter $\epsilon$. The problem possesses an irregular singularity in time at the origin and involves a set of so-called moving turning points merging to 0 with $\epsilon$. We construct outer solutions for time located in complex sectors that are kept away from the origin at a distance equivalent to a positive power of $|\epsilon|$ and we build up a related family of sectorial holomorphic inner solutions for small time inside some boundary layer. We show that both outer and inner solutions have Gevrey asymptotic expansions as $\epsilon$ tends to 0 on appropriate sets of sectors that cover a neighborhood of the origin in $ \mathbb{C}^{\ast}$. We observe that their Gevrey orders are distinct in general.
</p>projecteuclid.org/euclid.ade/1544497235_20181210220046Mon, 10 Dec 2018 22:00 ESTOn invariant measures associated with weakly coupled systems of Kolmogorov equationshttps://projecteuclid.org/euclid.ade/1548212468<strong>Davide Addona</strong>, <strong>Luciana Angiuli</strong>, <strong>Luca Lorenzi</strong>. <p><strong>Source: </strong>Advances in Differential Equations, Volume 24, Number 3/4, 137--184.</p><p><strong>Abstract:</strong><br/>
In this paper, we deal with weakly coupled elliptic systems ${\mathcal A}$ with unbounded coefficients. We prove the existence and characterize all the systems of invariant measures for the semigroup $({\bf T}(t))_{t\ge 0}$ associated with ${\mathcal A}$ in $C_b({\mathbb R^d};\mathbb R^m)$. We also show some relevant properties of the extension of $({\bf T}(t))_{t\ge 0}$ to the $L^p$-spaces related to systems of invariant measures. Finally, we study the asymptotic behaviour of $({\bf T}(t))_{t\ge 0}$ as $t$ tends to $+\infty$.
</p>projecteuclid.org/euclid.ade/1548212468_20190122220128Tue, 22 Jan 2019 22:01 ESTMultiplicity results for $(p,q)$ fractional elliptic equations involving critical nonlinearitieshttps://projecteuclid.org/euclid.ade/1548212469<strong>Mousomi Bhakta</strong>, <strong>Debangana Mukherjee</strong>. <p><strong>Source: </strong>Advances in Differential Equations, Volume 24, Number 3/4, 185--228.</p><p><strong>Abstract:</strong><br/>
In this paper, we prove the existence of infinitely many nontrivial solutions for the class of $ (p,q) $ fractional elliptic equations involving concave-critical nonlinearities in bounded domains in $\mathbb{R}^N$. Further, when the nonlinearity is of convex-critical type, we establish the multiplicity of nonnegative solutions using variational methods. In particular, we show the existence of at least $cat_{\Omega}(\Omega)$ nonnegative solutions.
</p>projecteuclid.org/euclid.ade/1548212469_20190122220128Tue, 22 Jan 2019 22:01 ESTGlobal existence and blowup for a quasilinear parabolic equations with nonlinear gradient absorptionhttps://projecteuclid.org/euclid.ade/1548212470<strong>Yingying Liu</strong>, <strong>Zhengce Zhang</strong>, <strong>Liping Zhu</strong>. <p><strong>Source: </strong>Advances in Differential Equations, Volume 24, Number 3/4, 229--256.</p><p><strong>Abstract:</strong><br/>
This paper deals with a quasilinear parabolic equation with nonlinear gradient absorption \begin{equation*} u_t-\Delta_{p}u=u^q-\mu u^{r}|\nabla u|^\delta, \ x\in\Omega, t>0. \end{equation*} Here, $\Delta_{p} u=\mathrm{div}(|\nabla u|^{p-2}\nabla u)$ is the p-Laplace operator, $\Omega \subset \mathbb{R}^{N}$ $(N \geq 1)$ is a bounded smooth domain. By a regularization approach, we first establish the local-in-time existence of its weak solutions. Then we prove the global existence by constructing a family of bounded super-solutions which technically depend on the inradius of $\Omega$. We also obtain an upper bound and a lower bound of the blowup time. We use a comparison with suitable self-similar sub-solutions to prove the blowup and an upper bound of blowup time. Finally, we derive a lower bound of the blowup time by using the differential inequality technique.
</p>projecteuclid.org/euclid.ade/1548212470_20190122220128Tue, 22 Jan 2019 22:01 EST