Showing papers in "Discrete and Continuous Dynamical Systems in 2022"

Global well-posedness for fractional Sobolev-Galpern type equations

Abstract: <p style='text-indent:20px;'>This article is a comparative study on an initial-boundary value problem for a class of semilinear pseudo-parabolic equations with the fractional Caputo derivative, also called the fractional Sobolev-Galpern type equations. The purpose of this work is to reveal the influence of the degree of the source nonlinearity on the well-posedness of the solution. By considering four different types of nonlinearities, we derive the global well-posedness of mild solutions to the problem corresponding to the four cases of the nonlinear source terms. For the advection source function case, we apply a nontrivial limit technique for singular integral and some appropriate choices of weighted Banach space to prove the global existence result. For the gradient nonlinearity as a local Lipschitzian, we use the Cauchy sequence technique to show that the solution either exists globally in time or blows up at finite time. For the polynomial form nonlinearity, by assuming the smallness of the initial data we derive the global well-posed results. And for the case of exponential nonlinearity in two-dimensional space, we derive the global well-posedness by additionally using an Orlicz space.</p>

Combined effects of singular and exponential nonlinearities in fractional Kirchhoff problems

Abstract: In this paper we establish the existence of at least two (weak) solutions for the following fractional Kirchhoff problem involving singular and exponential nonlinearities \begin{document}$ \begin{cases} M\left(\|u\|^{{n}/{s}}\right)(-\Delta)^s_{n/s}u = \mu u^{-q}+ u^{r-1}\exp( u^{\beta})\quad\text{in } \Omega,\\ u>0\qquad\text{in } \Omega,\\ u = 0\qquad\text{in } \mathbb R^n \setminus{ \Omega}, \end{cases} $\end{document} where \begin{document}$ \Omega $\end{document} is a smooth bounded domain of \begin{document}$ \mathbb R^n $\end{document} , \begin{document}$ n\geq 1 $\end{document} , \begin{document}$ s\in (0,1) $\end{document} , \begin{document}$ \mu>0 $\end{document} is a real parameter, \begin{document}$ \beta and \begin{document}$ q\in (0,1) $\end{document} .The paper covers the so called degenerate Kirchhoff case andthe existence proofs rely on the Nehari manifold techniques.

Combined effects of singular and exponential nonlinearities in fractional Kirchhoff problems

Abstract: <p style='text-indent:20px;'>In this paper we establish the existence of at least two (weak) solutions for the following fractional Kirchhoff problem involving singular and exponential nonlinearities</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{cases} M\left(\|u\|^{{n}/{s}}\right)(-\Delta)^s_{n/s}u = \mu u^{-q}+ u^{r-1}\exp( u^{\beta})\quad\text{in } \Omega,\\ u&gt;0\qquad\text{in } \Omega,\\ u = 0\qquad\text{in } \mathbb R^n \setminus{ \Omega}, \end{cases} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> is a smooth bounded domain of <inline-formula><tex-math id="M2">\begin{document}$ \mathbb R^n $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ n\geq 1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ s\in (0,1) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ \mu&gt;0 $\end{document}</tex-math></inline-formula> is a real parameter, <inline-formula><tex-math id="M6">\begin{document}$ \beta &lt;{n/(n-s)} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M7">\begin{document}$ q\in (0,1) $\end{document}</tex-math></inline-formula>.The paper covers the so called degenerate Kirchhoff case andthe existence proofs rely on the Nehari manifold techniques.</p>

Shadowing for families of endomorphisms of generalized group shifts

Abstract: Let \begin{document}$ G $\end{document} be a countable monoid and let \begin{document}$ A $\end{document} be an Artinian group (resp. an Artinian module). Let \begin{document}$ \Sigma \subset A^G $\end{document} be a closed subshift which is also a subgroup (resp. a submodule) of \begin{document}$ A^G $\end{document} . Suppose that \begin{document}$ \Gamma $\end{document} is a finitely generated monoid consisting of pairwise commuting cellular automata \begin{document}$ \Sigma \to \Sigma $\end{document} that are also homomorphisms of groups (resp. homomorphisms of modules) with monoid binary operation given by composition of maps. We show that the natural action of \begin{document}$ \Gamma $\end{document} on \begin{document}$ \Sigma $\end{document} satisfies a natural intrinsic shadowing property. Generalizations are also established for families of endomorphisms of admissible group subshifts.

A positive solution for an anisotropic $ (p,q) $-Laplacian

Abstract:

Here, the anisotropic \begin{document}$ (p, q) $\end{document}-Laplacian

\begin{document}$ - \sum\limits_{i = 1}^N\frac{\partial}{\partial x_i}\left( \left|\frac{\partial u}{\partial x_i}\right|^{p_i-2}\frac{\partial u}{\partial x_i}\right) - \sum\limits_{i = 1}^N\frac{\partial}{\partial x_i}\left( \left|\frac{\partial u}{\partial x_i}\right|^{q_i-2}\frac{\partial u}{\partial x_i}\right) = \lambda u^{\gamma-1} $\end{document}

is considered, where \begin{document}$ \Omega $\end{document} is a bounded and regular domain of \begin{document}$ \mathbb{R}^N $\end{document}, \begin{document}$ q_i\leq p_i $\end{document} for \begin{document}$ i = 1, \cdots, N $\end{document} and \begin{document}$ \gamma > 1 $\end{document}. The existence of positive solution is proved via sub-supersolution method.

Global well-posedness of the viscous Camassa–Holm equation with gradient noise

Abstract: . We analyse a nonlinear stochastic partial differential equation that corresponds to a viscous shallow water equation (of the Camassa–Holm type) perturbed by a convective, position-dependent noise term. We establish the existence of weak solutions in H m ( m ∈ N ) using Galerkin approximations and the stochastic compactness method. We derive a series of a priori esti- mates that combine a model-specific energy law with non-standard regularity estimates. We make systematic use of a stochastic Gronwall inequality and also stopping time techniques. The proof of convergence to a solution argues via tightness of the laws of the Galerkin solutions, and Skorokhod–Jakubowski a.s. representations of random variables in quasi-Polish spaces. The spatially dependent noise function constitutes a complication throughout the analysis, repeatedly giving rise to nonlinear terms that “balance” the martingale part of the equation against the second-order Stratonovich-to-Itˆo correction term. Fi- nally, via pathwise uniqueness, we conclude that the constructed solutions are probabilistically strong. The uniqueness proof is based on a finite-dimensional Itˆo formula and a DiPerna–Lions type regularisation procedure, where the regularisation errors are controlled by first and second order commutators. 60H15; 35A01, 35A02.

Cucker-Smale model with time delay

Abstract: <p style='text-indent:20px;'>We study the flocking model for continuous time introduced by Cucker and Smale adding a positive time delay <inline-formula><tex-math id="M1">\begin{document}$ \tau $\end{document}</tex-math></inline-formula>. The goal of this article is to prove that the same unconditional flocking result for the non-delayed case is valid in the delayed case. A novelty is that we do not need to impose any restriction on the size of <inline-formula><tex-math id="M2">\begin{document}$ \tau $\end{document}</tex-math></inline-formula>. Furthermore, when the unconditional flocking occurs, velocities converge exponentially fast to a common one.</p>

Shadowing for families of endomorphisms of generalized group shifts

Abstract: Let $G$ be a countable monoid and let $A$ be an Artinian group (resp. an Artinian module). Let $\Sigma \subset A^G$ be a closed subshift which is also a subgroup (resp. a submodule) of $A^G$. Suppose that $\Gamma$ is a finitely generated monoid consisting of pairwise commuting cellular automata $\Sigma \to \Sigma$ that are also homomorphisms of groups (resp. homomorphisms of modules) with monoid binary operation given by composition of maps. We show that the valuation action of $\Gamma$ on $\Sigma$ satisfies a natural intrinsic shadowing property. Generalizations are also established for families of endomorphisms of admissible group subshifts.

Eternal solutions for a reaction-diffusion equation with weighted reaction

Abstract: <p style='text-indent:20px;'>We prove existence and uniqueness of <i>eternal solutions</i> in self-similar form growing up in time with exponential rate for the weighted reaction-diffusion equation</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \partial_tu = \Delta u^m+|x|^{\sigma}u^p, $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>posed in <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^N $\end{document}</tex-math></inline-formula>, with <inline-formula><tex-math id="M2">\begin{document}$ m&gt;1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ 0&lt;p&lt;1 $\end{document}</tex-math></inline-formula> and the critical value for the weight</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \sigma = \frac{2(1-p)}{m-1}. $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>Existence and uniqueness of some specific solution holds true when <inline-formula><tex-math id="M4">\begin{document}$ m+p\geq2 $\end{document}</tex-math></inline-formula>. On the contrary, no eternal solution exists if <inline-formula><tex-math id="M5">\begin{document}$ m+p&lt;2 $\end{document}</tex-math></inline-formula>. We also classify exponential self-similar solutions with a different interface behavior when <inline-formula><tex-math id="M6">\begin{document}$ m+p&gt;2 $\end{document}</tex-math></inline-formula>. Some transformations to reaction-convection-diffusion equations and traveling wave solutions are also introduced.</p>

Global existence and convergence to steady states for a predator-prey model with both predator- and prey-taxis

Abstract: <p style='text-indent:20px;'>In this work we consider a two-species predator-prey chemotaxis model</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \left\{ \begin{array}{lll} u_t = d_1\Delta u+\chi_1 abla\cdot(u abla v)+u(a_1-b_{11}u-b_{12}v), &amp;x\in \Omega, t&gt;0, \\[0.2cm] v_t = d_2\Delta v-\chi_2 abla\cdot(v abla u)+v(-a_2+b_{21}u-b_{22}v-b_{23}w), &amp; x\in \Omega, t&gt;0, \\[0.2cm] w_t = d_3\Delta w-\chi_3 abla\cdot(w abla v)+w(-a_3+b_{32}v-b_{33}w), &amp; x\in \Omega, t&gt;0 \\ \end{array}\right.(\ast) $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>in a bounded domain with smooth boundary. We prove that if (1.7)-(1.13) hold, the model (<inline-formula><tex-math id="M1">\begin{document}$ \ast $\end{document}</tex-math></inline-formula>) admits a global boundedness of classical solutions in any physically meaningful dimension. Moreover, we show that the global classical solutions <inline-formula><tex-math id="M2">\begin{document}$ (u,v,w) $\end{document}</tex-math></inline-formula> exponentially converges to constant stable steady state <inline-formula><tex-math id="M3">\begin{document}$ (u_\ast,v_\ast,w_\ast) $\end{document}</tex-math></inline-formula>. Inspired by [<xref ref-type="bibr" rid="b5">5</xref>], we employ the special structure of (<inline-formula><tex-math id="M4">\begin{document}$ \ast $\end{document}</tex-math></inline-formula>) and carefully balance the triple cross diffusion. Indeed, we introduced some functions and combined them in a way that allowed us to cancel the bad items.</p>

The Brinkman-Fourier system with ideal gas equilibrium

Abstract: <p style='text-indent:20px;'>In this work, we will introduce a general framework to derive the thermodynamics of a fluid mechanical system, which guarantees the consistence between the energetic variational approaches with the laws of thermodynamics. In particular, we will focus on the coupling between the thermal and mechanical forces. We follow the framework for a classical gas with ideal gas equilibrium and present the existences of weak solutions to this thermodynamic system coupled with the Brinkman-type equation to govern the velocity field.</p>

Ground states for a system of nonlinear Schrödinger equations with singular potentials

Abstract: In this paper, we consider the existence and asymptotic behavior of ground state solutions for a class of Hamiltonian elliptic system with Hardy potential. The resulting problem engages three major difficulties: one is that the associated functional is strongly indefinite, the second difficulty we must overcome lies in verifying the link geometry and showing the boundedness of Cerami sequences when the nonlinearity is different from the usual global super-quadratic condition. The third difficulty is singular potential, which does not belong to the Kato's class. These enable us to develop a direct approach and new tricks to overcome the difficulties caused by singularity of potential and the dropping of classical super-quadratic assumption on the nonlinearity. Our approach is based on non-Nehari method which developed recently, we establish some new existence results of ground state solutions of Nehari-Pankov type under some mild conditions, and analyze asymptotical behavior of ground state solutions.

Fujita type results for quasilinear parabolic inequalities with nonlocal terms

Abstract: <p style='text-indent:20px;'>In this paper we investigate the nonexistence of nonnegative solutions of parabolic inequalities of the form</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{cases} &amp;u_t \pm L_\mathcal A u\geq (K\ast u^p)u^q \quad\mbox{ in } \mathbb R^N \times \mathbb (0,\infty),\, N\geq 1,\\ &amp;u(x,0) = u_0(x)\ge0 \,\, \text{ in } \mathbb R^N,\end{cases} \qquad (P^{\pm}) $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ u_0\in L^1_{loc}({\mathbb R}^N) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ L_{\mathcal{A}} $\end{document}</tex-math></inline-formula> denotes a weakly <inline-formula><tex-math id="M3">\begin{document}$ m $\end{document}</tex-math></inline-formula>-coercive operator, which includes as prototype the <inline-formula><tex-math id="M4">\begin{document}$ m $\end{document}</tex-math></inline-formula>-Laplacian or the generalized mean curvature operator, <inline-formula><tex-math id="M5">\begin{document}$ p,\,q&gt;0 $\end{document}</tex-math></inline-formula>, while <inline-formula><tex-math id="M6">\begin{document}$ K\ast u^p $\end{document}</tex-math></inline-formula> stands for the standard convolution operator between a weight <inline-formula><tex-math id="M7">\begin{document}$ K&gt;0 $\end{document}</tex-math></inline-formula> satisfying suitable conditions at infinity and <inline-formula><tex-math id="M8">\begin{document}$ u^p $\end{document}</tex-math></inline-formula>. For problem <inline-formula><tex-math id="M9">\begin{document}$ (P^-) $\end{document}</tex-math></inline-formula> we obtain a Fujita type exponent while for <inline-formula><tex-math id="M10">\begin{document}$ (P^+) $\end{document}</tex-math></inline-formula> we show that no such critical exponent exists. Our approach relies on nonlinear capacity estimates adapted to the nonlocal setting of our problems. No comparison results or maximum principles are required.</p>

Lower bounds for Orlicz eigenvalues

Abstract: <p style='text-indent:20px;'>In this article we consider the following weighted nonlinear eigenvalue problem for the <inline-formula><tex-math id="M1">\begin{document}$ g- $\end{document}</tex-math></inline-formula>Laplacian</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ -{\text{ div}}\left( g(| abla u|)\frac{ abla u}{| abla u|}\right) = \lambda w(x) h(|u|)\frac{u}{|u|} \quad \text{ in }\Omega\subset \mathbb R^n, n\geq 1 $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>with Dirichlet boundary conditions. Here <inline-formula><tex-math id="M2">\begin{document}$ w $\end{document}</tex-math></inline-formula> is a suitable weight and <inline-formula><tex-math id="M3">\begin{document}$ g = G' $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ h = H' $\end{document}</tex-math></inline-formula> are appropriated Young functions satisfying the so called <inline-formula><tex-math id="M5">\begin{document}$ \Delta' $\end{document}</tex-math></inline-formula> condition, which includes for instance logarithmic perturbation of powers and different power behaviors near zero and infinity. We prove several properties on its spectrum, being our main goal to obtain lower bounds of eigenvalues in terms of <inline-formula><tex-math id="M6">\begin{document}$ G $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M7">\begin{document}$ H $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M8">\begin{document}$ w $\end{document}</tex-math></inline-formula> and the normalization <inline-formula><tex-math id="M9">\begin{document}$ \mu $\end{document}</tex-math></inline-formula> of the corresponding eigenfunctions.</p><p style='text-indent:20px;'>We introduce some new strategies to obtain results that generalize several inequalities from the literature of <inline-formula><tex-math id="M10">\begin{document}$ p- $\end{document}</tex-math></inline-formula>Laplacian type eigenvalues.</p>

The nonlocal-interaction equation near attracting manifolds

Abstract: <p style='text-indent:20px;'>We study the approximation of the nonlocal-interaction equation restricted to a compact manifold <inline-formula><tex-math id="M1">\begin{document}$ {\mathcal{M}} $\end{document}</tex-math></inline-formula> embedded in <inline-formula><tex-math id="M2">\begin{document}$ {\mathbb{R}}^d $\end{document}</tex-math></inline-formula>, and more generally compact sets with positive reach (i.e. prox-regular sets). We show that the equation on <inline-formula><tex-math id="M3">\begin{document}$ {\mathcal{M}} $\end{document}</tex-math></inline-formula> can be approximated by the classical nonlocal-interaction equation on <inline-formula><tex-math id="M4">\begin{document}$ {\mathbb{R}}^d $\end{document}</tex-math></inline-formula> by adding an external potential which strongly attracts to <inline-formula><tex-math id="M5">\begin{document}$ {\mathcal{M}} $\end{document}</tex-math></inline-formula>. The proof relies on the Sandier–Serfaty approach [<xref ref-type="bibr" rid="b23">23</xref>,<xref ref-type="bibr" rid="b24">24</xref>] to the <inline-formula><tex-math id="M6">\begin{document}$ \Gamma $\end{document}</tex-math></inline-formula>-convergence of gradient flows. As a by-product, we recover well-posedness for the nonlocal-interaction equation on <inline-formula><tex-math id="M7">\begin{document}$ {\mathcal{M}} $\end{document}</tex-math></inline-formula>, which was shown [<xref ref-type="bibr" rid="b10">10</xref>]. We also provide an another approximation to the interaction equation on <inline-formula><tex-math id="M8">\begin{document}$ {\mathcal{M}} $\end{document}</tex-math></inline-formula>, based on iterating approximately solving an interaction equation on <inline-formula><tex-math id="M9">\begin{document}$ {\mathbb{R}}^d $\end{document}</tex-math></inline-formula> and projecting to <inline-formula><tex-math id="M10">\begin{document}$ {\mathcal{M}} $\end{document}</tex-math></inline-formula>. We show convergence of this scheme, together with an estimate on the rate of convergence. Finally, we conduct numerical experiments, for both the attractive-potential-based and the projection-based approaches, that highlight the effects of the geometry on the dynamics.</p>

One component regularity criteria for the axially symmetric MHD-Boussinesq system

Abstract: In this paper, we consider regularity criteria of a class of 3D axially symmetric MHD-Boussinesq systems without magnetic resistivity or thermal diffusivity. Under some Prodi-Serrin type critical assumptions on the horizontal angular component of the velocity, we will prove that strong solutions of the axially symmetric MHD-Boussinesq system can be smoothly extended beyond the possible blow-up time $T_\ast$ if the magnetic field contains only the horizontal swirl component. No a priori assumption on the magnetic field and the temperature fluctuation is imposed.

On the Dirichlet problem for Lagrangian phase equation with critical and supercritical phase

Abstract: In this paper, we solve the Dirichlet problem for Lagrangian phase equation with critical and supercritical phase. A crucial ingredient is the interior $C^2$ estimate. Our result is sharp in the sense that there exist singular solutions in the subcritical phase case.

Polynomial ergodic averages for certain countable ring actions

Abstract: <p style='text-indent:20px;'>A recent result of Frantzikinakis in [<xref ref-type="bibr" rid="b17">17</xref>] establishes sufficient conditions for joint ergodicity in the setting of <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{Z} $\end{document}</tex-math></inline-formula>-actions. We generalize this result for actions of second-countable locally compact abelian groups. We obtain two applications of this result. First, we show that, given an ergodic action <inline-formula><tex-math id="M2">\begin{document}$ (T_n)_{n \in F} $\end{document}</tex-math></inline-formula> of a countable field <inline-formula><tex-math id="M3">\begin{document}$ F $\end{document}</tex-math></inline-formula> with characteristic zero on a probability space <inline-formula><tex-math id="M4">\begin{document}$ (X,\mathcal{B},\mu) $\end{document}</tex-math></inline-formula> and a family <inline-formula><tex-math id="M5">\begin{document}$ \{p_1,\dots,p_k\} $\end{document}</tex-math></inline-formula> of independent polynomials, we have</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \lim\limits_{N \to \infty} \frac{1}{|\Phi_N|}\sum\limits_{n \in \Phi_N} T_{p_1(n)}f_1\cdots T_{p_k(n)}f_k\ = \ \prod\limits_{j = 1}^k \int_X f_i \ d\mu, $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M6">\begin{document}$ f_i \in L^{\infty}(\mu) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M7">\begin{document}$ (\Phi_N) $\end{document}</tex-math></inline-formula> is a Følner sequence of <inline-formula><tex-math id="M8">\begin{document}$ (F,+) $\end{document}</tex-math></inline-formula>, and the convergence takes place in <inline-formula><tex-math id="M9">\begin{document}$ L^2(\mu) $\end{document}</tex-math></inline-formula>. This yields corollaries in combinatorics and topological dynamics. Second, we prove that a similar result holds for totally ergodic actions of suitable rings.</p>

Stability of hyperbolic Oseledets splittings for quasi-compact operator cocycles

Abstract: <p style='text-indent:20px;'>We develop a random version of the perturbation theory of Gouëzel, Keller, and Liverani, and consequently obtain results on the stability of Oseledets splittings and Lyapunov exponents for operator cocycles. By applying the theory to the Perron-Frobenius operator cocycles associated to random <inline-formula><tex-math id="M1">\begin{document}$ \mathcal{C}^k $\end{document}</tex-math></inline-formula> expanding maps on <inline-formula><tex-math id="M2">\begin{document}$ S^1 $\end{document}</tex-math></inline-formula> (<inline-formula><tex-math id="M3">\begin{document}$ k \ge 2 $\end{document}</tex-math></inline-formula>) we provide conditions for the stability of Lyapunov exponents and Oseledets splitting of the cocycle under (ⅰ) uniformly small fiber-wise <inline-formula><tex-math id="M4">\begin{document}$ \mathcal{C}^{k-1} $\end{document}</tex-math></inline-formula>-perturbations to the random dynamics, and (ⅱ) numerical approximation via a Fejér kernel method. A notable addition to our approach is the use of Saks spaces, which allow us to weaken the hypotheses of Gouëzel-Keller-Liverani perturbation theory, provides a unifying framework for key concepts in the so-called 'functional analytic' approach to studying dynamical systems, and has applications to the construction of anisotropic norms adapted to dynamical systems.</p>

Existence of solutions for fractional instantaneous and non-instantaneous impulsive differential equations with perturbation and Dirichlet boundary value

Abstract: A class of fractional instantaneous and non-instantaneous impulsive differential equations under Dirichlet boundary value conditions with perturbation is considered here. The existence of classical solutions is presented by using the Weierstrass theorem. An example is given to verify the validity of the obtained results.

The Brinkman-Fourier system with ideal gas equilibrium

Abstract: In this work, we will introduce a general framework to derive the thermodynamics of a fluid mechanical system, which guarantees the consistence between the energetic variational approaches with the laws of thermodynamics. In particular, we will focus on the coupling between the thermal and mechanical forces. We follow the framework for a classical gas with ideal gas equilibrium and present the existences of weak solutions to this thermodynamic system coupled with the Brinkman-type equation to govern the velocity field.

Criniferous entire maps with absorbing Cantor bouquets

Abstract: <p style='text-indent:20px;'>It is known that, for many transcendental entire functions in the Eremenko-Lyubich class <inline-formula><tex-math id="M1">\begin{document}$ \mathcal{B} $\end{document}</tex-math></inline-formula>, every escaping point can eventually be connected to infinity by a curve of escaping points. When this is the case, we say that the functions are <i>criniferous</i>. In this paper, we extend this result to a new class of maps in <inline-formula><tex-math id="M2">\begin{document}$ \mathcal{B} $\end{document}</tex-math></inline-formula>. Furthermore, we show that if a map belongs to this class, then its Julia set contains a <i>Cantor bouquet</i>; in other words, it is a subset of <inline-formula><tex-math id="M3">\begin{document}$ \mathbb{C} $\end{document}</tex-math></inline-formula> ambiently homeomorphic to a straight brush.</p>

Higher order parabolic boundary Harnack inequality in C1 and Ck, α domains

Abstract:

We study the boundary behaviour of solutions to second order parabolic linear equations in moving domains. Our main result is a higher order boundary Harnack inequality in C¹ and C^{k, α} domains, providing that the quotient of two solutions vanishing on the boundary of the domain is as smooth as the boundary.

As a consequence of our result, we provide a new proof of higher order regularity of the free boundary in the parabolic obstacle problem.

On the improved interior regularity of a boundary value problem modelling the displacement of a linearly elastic elliptic membrane shell subject to an obstacle

Abstract: <p style='text-indent:20px;'>In this paper we show that the solution of an obstacle problem for linearly elastic elliptic membrane shells enjoys higher differentiability properties in the interior of the domain where it is defined.</p>

Stabilization of periodic sweeping processes and asymptotic average velocity for soft locomotors with dry friction

Abstract: <p style='text-indent:20px;'>We study the asymptotic stability of periodic solutions for sweeping processes defined by a polyhedron with translationally moving faces. Previous results are improved by obtaining a stronger <inline-formula><tex-math id="M1">\begin{document}$ W^{1,2} $\end{document}</tex-math></inline-formula> convergence. Then we present an application to a model of crawling locomotion. Our stronger convergence allows us to prove the stabilization of the system to a running-periodic (or derivo-periodic, or relative-periodic) solution and the well-posedness of an average asymptotic velocity depending only on the gait adopted by the crawler. Finally, we discuss some examples of finite-time versus asymptotic-only convergence.</p>

Multiple ergodic averages for variable polynomials

Abstract: <p style='text-indent:20px;'>In this paper we study multiple ergodic averages for "good" variable polynomials. In particular, under an additional assumption, we show that these averages converge to the expected limit, making progress related to an open problem posted by Frantzikinakis ([<xref ref-type="bibr" rid="b13">13</xref>,Problem 10]). These general convergence results imply several variable extensions of classical recurrence, combinatorial and number theoretical results which are presented as well.</p>

Upper and lower $ L^2 $-decay bounds for a class of derivative nonlinear Schrödinger equations

Abstract:

We consider the initial value problem for cubic derivative nonlinear Schrödinger equations possessing weakly dissipative structure in one space dimension. We show that the small data solution decays like \begin{document}$ O((\log t)^{-1/4}) $\end{document} in \begin{document}$ L^2 $\end{document} as \begin{document}$ t\to +\infty $\end{document}. Furthermore, we find that this \begin{document}$ L^2 $\end{document}-decay rate is optimal by giving a lower estimate of the same order.

Emergence of biological transportation networks as a self-regulated process

Abstract: We study self-regulating processes modeling biological transportation networks. Firstly, we write the formal $L^2$-gradient flow for the symmetric tensor valued diffusivity $D$ of a broad class of entropy dissipations associated with a purely diffusive model. The introduction of a prescribed electric potential leads to the Fokker-Planck equation, for whose entropy dissipations we also investigate the formal $L^2$-gradient flow. We derive an integral formula for the second variation of the dissipation functional, proving convexity (in dependence of diffusivity tensor) for a quadratic entropy density modeling Joule heating. Finally, we couple in the Poisson equation for the electric potential obtaining the Poisson-Nernst-Planck system. The formal gradient flow of the associated entropy loss functional is derived, giving an evolution equation for $D$ coupled with two auxiliary elliptic PDEs.

Random attractors for dissipative systems with rough noises

Abstract: <p style='text-indent:20px;'>We provide an analytic approach to study the asymptotic dynamics of rough differential equations, with the driving noises of Hölder continuity. Such systems can be solved with Lyons' theory of rough paths, in particular the rough integrals are understood in the Gubinelli sense for controlled rough paths. Using the framework of random dynamical systems and random attractors, we prove the existence and upper semi-continuity of the global pullback attractor for dissipative systems perturbed by bounded noises. Moreover, if the unperturbed system is strictly dissipative then the random attractor is a singleton for sufficiently small noise intensity.</p>

Parabolic methods for ultraspherical interpolation inequalities

Abstract: The carré du champ method is a powerful technique for proving interpolation inequalities with explicit constants in presence of a non-trivial metric on a manifold. The method applies to some classical Gagliardo-Nirenberg-Sobolev inequalities on the sphere, with optimal constants. Very nonlinear regimes close to the critical Sobolev exponent can be covered using nonlinear parabolic flows of porous medium or fast diffusion type. Considering power law weights is a natural question in relation with symmetry breaking issues for Caffarelli-Kohn-Nirenberg inequalities, but regularity estimates for a complete justification of the computation are missing. We provide the first example of a complete parabolic proof based on a nonlinear flow by regularizing the singularity induced by the weight. Our result is established in the simplified framework of a diffusion built on the ultraspherical operator, which amounts to reduce the problem to functions on the sphere with simple symmetry properties.