# American Institute of Mathematical Sciences

ISSN:
1078-0947

eISSN:
1553-5231

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## Discrete & Continuous Dynamical Systems

June 2021 , Volume 41 , Issue 6

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2021, 41(6): 2519-2542 doi: 10.3934/dcds.2020373 +[Abstract](597) +[HTML](195) +[PDF](402.08KB)
Abstract:

A distributed optimal control problem for a diffuse interface model, which physical context is that of tumour growth dynamics, is addressed. The system we deal with comprises a Cahn–Hilliard equation for the tumour fraction coupled with a reaction-diffusion for a nutrient species surrounding the tumourous cells. The cost functional to be minimised possesses some objective terms and it also penalises long treatments time, which may affect harm to the patients, and big aggregations of tumourous cells. Hence, the optimisation problem leads to the optimal strategy which reduces the time exposure of the patient to the medication and at the same time allows the doctors to achieve suitable clinical goals.

2021, 41(6): 2543-2557 doi: 10.3934/dcds.2020374 +[Abstract](561) +[HTML](184) +[PDF](304.68KB)
Abstract:

The present work is devoted to study comparison and converse comparison theorems for diagonally quadratic BSDEs. We give sufficient and necessary conditions under which the comparison holds. Sufficient and necessary conditions for non-positive and non-negative solutions are presented.

2021, 41(6): 2559-2599 doi: 10.3934/dcds.2020375 +[Abstract](526) +[HTML](189) +[PDF](555.91KB)
Abstract:

We study geometric and algebraic geometric properties of the continuous and discrete Neumann systems on cotangent bundles of Stiefel varieties \begin{document}$V_{n,r}$\end{document}. The systems are integrable in the non-commutative sense, and by applying a \begin{document}$2r\times 2r$\end{document}–Lax representation, we show that generic complex invariant manifolds are open subsets of affine Prym varieties on which the complex flow is linear. The characteristics of the varieties and the direction of the flow are calculated explicitly. Next, we construct a family of multi-valued integrable discretizations of the Neumann systems and describe them as translations on the Prym varieties, which are written explicitly in terms of divisors of points on the spectral curve.

2021, 41(6): 2601-2617 doi: 10.3934/dcds.2020376 +[Abstract](597) +[HTML](196) +[PDF](372.69KB)
Abstract:

The parabolic-parabolic Keller-Segel model of chemotaxis is shown to come up as the hydrodynamic system describing the evolution of the modulus square \begin{document}$n(t,x)$\end{document} and the argument \begin{document}$S(t,x)$\end{document} of a wavefunction \begin{document}$\psi = \sqrt{n} \, e^{iS}$\end{document} that solves a cubic Schrödinger equation with focusing interaction, frictional Kostin nonlinearity and Doebner-Goldin dissipation mechanism. This connection is then exploited to construct a family of quasi-stationary solutions to the Keller-Segel system under the influence of no-flux and anti-Fick laws.

2021, 41(6): 2619-2633 doi: 10.3934/dcds.2020377 +[Abstract](597) +[HTML](195) +[PDF](336.65KB)
Abstract:

We derive second order estimates for \begin{document}$\chi$\end{document}-plurisubharmonic solutions of complex Hessian equations with right hand side depending on the gradient on compact Hermitian manifolds.

Lei Liu and
2021, 41(6): 2635-3652 doi: 10.3934/dcds.2020378 +[Abstract](582) +[HTML](187) +[PDF](412.16KB)
Abstract:

There is a long standing conjecture that there are at least \begin{document}$n$\end{document} closed characteristics on any compact convex hypersurface \begin{document}$\Sigma$\end{document} in \begin{document}$\mathbb{R}^{2n}$\end{document}. In this paper, we provide some new estimates and prove that there are at least \begin{document}$[\frac{3n}{4}]$\end{document} closed characteristics on \begin{document}$\Sigma$\end{document} for any positive integer \begin{document}$n$\end{document}, where \begin{document}$\Sigma$\end{document} satisfies \begin{document}$\Sigma = P\Sigma$\end{document} for a certain class of symplectic matrix \begin{document}$P$\end{document}. These results are not considered in previous papers.

2021, 41(6): 2653-2676 doi: 10.3934/dcds.2020379 +[Abstract](638) +[HTML](218) +[PDF](457.01KB)
Abstract:

In this paper, we study the following coupled nonlocal system

where \begin{document}$(-\Delta)^{s}$\end{document} is the fractional Laplacian, \begin{document}$0<s<1$\end{document}, \begin{document}$\mu_{1},\, \mu_{2}>0$\end{document}, \begin{document}$N>2s$\end{document}, and \begin{document}$\frac{4s}{N}<\alpha\leq \frac{2s}{N-2s}$\end{document}. We are concerned with the attractive case, which corresponds to \begin{document}$\beta>0$\end{document}. In the case of low perturbations of the coupling parameter, by using two-dimensional linking arguments, we show that there exists \begin{document}$\beta_{1}>0$\end{document} such that when \begin{document}$0<\beta<\beta_{1}$\end{document}, then the system has a positive radial solution. Next, in the case of high perturbations of the coupling parameter, we prove that there exists \begin{document}$\beta_{2}>0$\end{document} such that the system has a mountain-pass type solution for all \begin{document}$\beta>\beta_{2}$\end{document}. These results correspond to low and high perturbations with respect to the values of the coupling parameter \begin{document}$\beta$\end{document}. This paper extends and complements the main results established in [2] for the particular case \begin{document}$N = 3$\end{document}, \begin{document}$s = 1$\end{document}, \begin{document}$\alpha = 2$\end{document}.

2021, 41(6): 2677-2698 doi: 10.3934/dcds.2020381 +[Abstract](486) +[HTML](187) +[PDF](463.13KB)
Abstract:

In this paper we study flows having an isolated non-saddle set. We see that the global structure of a flow having an isolated non-saddle set \begin{document}$K$\end{document} depends on the way \begin{document}$K$\end{document} sits in the phase space at the cohomological level. We construct flows on surfaces having isolated non-saddle sets with a prescribed global structure. We also study smooth parametrized families of flows and continuations of isolated non-saddle sets.

2021, 41(6): 2699-2723 doi: 10.3934/dcds.2020382 +[Abstract](781) +[HTML](180) +[PDF](1035.3KB)
Abstract:

We consider the initial value problem associated to a coupled system of modified Korteweg-de Vries type equations

and prove the local well-posedness results for a given data in low regularity Sobolev spaces \begin{document}$H^{s}( \rm{I}\! \rm{R})\times H^{k}( \rm{I}\! \rm{R})$\end{document}, \begin{document}$s,k> -\frac12$\end{document} and \begin{document}$|s-k|\leq 1/2$\end{document}, for \begin{document}$\alpha\neq 0,1$\end{document}. Also, we prove that: (I) the solution mapping that takes initial data to the solution fails to be \begin{document}$C^3$\end{document} at the origin, when \begin{document}$s<-1/2$\end{document} or \begin{document}$k<-1/2$\end{document} or \begin{document}$|s-k|>2$\end{document}; (II) the trilinear estimates used in the proof of the local well-posedness theorem fail to hold when (a) \begin{document}$s-2k>1$\end{document} or \begin{document}$k<-1/2$\end{document} (b) \begin{document}$k-2s>1$\end{document} or \begin{document}$s<-1/2$\end{document}; (c) \begin{document}$s = k = -1/2$\end{document};

2021, 41(6): 2725-3737 doi: 10.3934/dcds.2020383 +[Abstract](626) +[HTML](210) +[PDF](366.36KB)
Abstract:

In this paper, we investigate the non-autonomous stochastic evolution equations of parabolic type with nonlinear noise and nonlocal initial conditions in Hilbert spaces, where the operators in linear part depend on time \begin{document}$t$\end{document} and generate an noncompact evolution family. New existence result of mild solutions is established under some weaker growth and measure of noncompactness conditions on nonlinear functions and nonlocal term. The discussions are based on Sadovskii's fixed-point theorem as well as the theory of evolution family. At last, as a sample of application, the obtained abstract result is applied to a class of non-autonomous stochastic partial differential equations of parabolic type with nonlocal initial conditions. The result obtained in this paper is a supplement to the existing literature and essentially extends some existing results in this area.

2021, 41(6): 2739-2776 doi: 10.3934/dcds.2020384 +[Abstract](529) +[HTML](177) +[PDF](666.62KB)
Abstract:

We introduce the notion of forward untangled Lagrangian representation of a measure-divergence vector-measure \begin{document}$\rho(1, {\mathit{\boldsymbol{b}}})$\end{document}, where \begin{document}$\rho \in \mathcal{M}^+( \mathbb{R}^{d+1})$\end{document} and \begin{document}${\mathit{\boldsymbol{b}}} \colon \mathbb{R}^{d+1} \to \mathbb{R}^d$\end{document} is a \begin{document}$\rho$\end{document}-integrable vector field with \begin{document}${\rm{div}}_{t,x}(\rho(1, {\mathit{\boldsymbol{b}}})) = \mu \in \mathcal M( \mathbb{R} \times \mathbb{R}^d)$\end{document}: forward untangling formalizes the notion of forward uniqueness in the language of Lagrangian representations. We identify local conditions for a Lagrangian representation to be forward untangled, and we show how to derive global forward untangling from such local assumptions. We then show how to reduce the PDE \begin{document}${\rm{div}}_{t,x}(\rho(1, {\mathit{\boldsymbol{b}}})) = \mu$\end{document} on a partition of \begin{document}$\mathbb{R}^+ \times \mathbb{R}^d$\end{document} obtained concatenating the curves seen by the Lagrangian representation. As an application, we recover known well posedeness results for the flow of monotone vector fields and for the associated continuity equation.

2021, 41(6): 2777-2808 doi: 10.3934/dcds.2020385 +[Abstract](545) +[HTML](174) +[PDF](408.57KB)
Abstract:

In this paper we consider the homogenization problem for a nonlocal equation that involve different smooth kernels. We assume that the spacial domain is divided into a sequence of two subdomains \begin{document}$A_n \cup B_n$\end{document} and we have three different smooth kernels, one that controls the jumps from \begin{document}$A_n$\end{document} to \begin{document}$A_n$\end{document}, a second one that controls the jumps from \begin{document}$B_n$\end{document} to \begin{document}$B_n$\end{document} and the third one that governs the interactions between \begin{document}$A_n$\end{document} and \begin{document}$B_n$\end{document}. Assuming that \begin{document}$\chi_{A_n} (x) \to X(x)$\end{document} weakly-* in \begin{document}$L^\infty$\end{document} (and then \begin{document}$\chi_{B_n} (x) \to (1-X)(x)$\end{document} weakly-* in \begin{document}$L^\infty$\end{document}) as \begin{document}$n \to \infty$\end{document} we show that there is an homogenized limit system in which the three kernels and the limit function \begin{document}$X$\end{document} appear. We deal with both Neumann and Dirichlet boundary conditions. Moreover, we also provide a probabilistic interpretation of our results.

2021, 41(6): 2809-2828 doi: 10.3934/dcds.2020386 +[Abstract](486) +[HTML](188) +[PDF](458.28KB)
Abstract:

We prove that given a measure preserving system \begin{document}$(X,\mathcal{B},\mu,T_1,\dots, T_d)$\end{document} with commuting, ergodic transformations \begin{document}$T_i$\end{document} such that \begin{document}$T_iT_j^{-1}$\end{document} are ergodic for all \begin{document}$i \neq j$\end{document}, the multicorrelation sequence \begin{document}$a(n) = \int_X f_0 \cdot T_1^nf_1 \cdot \dotso \cdot T_d^n f_d \ d\mu$\end{document} can be decomposed as \begin{document}$a(n) = a_{ \rm{st}}(n)+a_{ \rm{er}}(n)$\end{document}, where \begin{document}$a_{ \rm{st}}$\end{document} is a uniform limit of \begin{document}$d$\end{document}-step nilsequences and \begin{document}$a_{ \rm{er}}$\end{document} is a nullsequence (that is, \begin{document}$\lim_{N-M \to \infty} \frac{1}{N-M} \sum_{n = M}^{N-1} |a_{ \rm{er}}|^2 = 0$\end{document}). Under some additional ergodicity conditions on \begin{document}$T_1,\dots,T_d$\end{document} we also establish a similar decomposition for polynomial multicorrelation sequences of the form \begin{document}$a(n) = \int_X f_0 \cdot \prod_{i = 1}^dT_i^{p_{i,1}(n)}f_1\cdot\dotso \cdot \prod_{i = 1}^dT_i^{p_{i,k}(n)}f_k \ d\mu$\end{document}, where each \begin{document}$p_{i,k}: {\mathbb{Z}} \rightarrow {\mathbb{Z}}$\end{document} is a polynomial map. We also show, for \begin{document}$d = 2$\end{document}, that if \begin{document}$T_1, T_2, T_1T_2^{-1}$\end{document} are invertible and ergodic, we have large triple intersections: for all \begin{document}$\varepsilon>0$\end{document} and all \begin{document}$A \in \mathcal{B}$\end{document}, the set \begin{document}$\{n \in {\mathbb{Z}} : \mu(A \cap T_1^{-n}A \cap T_2^{-n}A)>\mu(A)^3-\varepsilon\}$\end{document} is syndetic. Moreover, we show that if \begin{document}$T_1, T_2, T_1T_2^{-1}$\end{document} are totally ergodic, and we denote by \begin{document}$p_n$\end{document} the \begin{document}$n$\end{document}-th prime, the set \begin{document}$\{n \in \mathbb{N} : \mu(A \cap T_1^{-(p_n-1)}A \cap T_2^{-(p_n-1)}A)>\mu(A)^3-\varepsilon\}$\end{document} has positive lower density.

2021, 41(6): 2829-2871 doi: 10.3934/dcds.2020388 +[Abstract](581) +[HTML](145) +[PDF](997.13KB)
Abstract:

First, we prove existence, nonnegativity, and pathwise uniqueness of martingale solutions to stochastic porous-medium equations driven by conservative multiplicative power-law noise in the Ito-sense. We rely on an energy approach based on finite-element discretization in space, homogeneity arguments and stochastic compactness. Secondly, we use Monte-Carlo simulations to investigate the impact noise has on waiting times and on free-boundary propagation. We find strong evidence that noise on average significantly accelerates propagation and reduces the size of waiting times – changing in particular scaling laws for the size of waiting times.

2021, 41(6): 2873-2890 doi: 10.3934/dcds.2020389 +[Abstract](535) +[HTML](141) +[PDF](367.46KB)
Abstract:

For \begin{document}$x\in[0,1),$\end{document} let \begin{document}$[d_{1}(x),d_{2}(x),\ldots]$\end{document} be its Lüroth expansion and \begin{document}$\big\{\frac{p_{n}(x)}{q_{n}(x)}, n\geq 1\big\}$\end{document} be the sequence of convergents of \begin{document}$x.$\end{document} In this paper, we study the Jarník-like set of real numbers which can be well approximated by infinitely many of their convergents in the Lüroth expansion\begin{document}$\colon$\end{document}

where \begin{document}$\psi\colon \mathbb{R}\to (0,\frac{1}{2}]$\end{document} is a positive function. We completely determine the Hausdorff dimension of \begin{document}$W(\psi).$\end{document}

2021, 41(6): 2891-2905 doi: 10.3934/dcds.2020390 +[Abstract](471) +[HTML](150) +[PDF](354.21KB)
Abstract:

In this note we consider a symmetric Random Walk defined by a \begin{document}$(f, f^{-1})$\end{document} Kalikow type system, where \begin{document}$f$\end{document} is the time-one map of the geodesic flow corresponding to an hyperbolic manifold. We provide necessary and sufficient conditions for the existence of an stationary measure for the walk that is equivalent to the volume in the corresponding unit tangent bundle. Some dynamical consequences for the Random Walk are deduced in these cases.

2021, 41(6): 2907-2946 doi: 10.3934/dcds.2020391 +[Abstract](420) +[HTML](154) +[PDF](491.07KB)
Abstract:

We consider an optimization problem with volume constraint for an energy functional associated to an inhomogeneous operator with nonstandard growth. By studying an auxiliary penalized problem, we prove existence and regularity of solution to the original problem: every optimal configuration is a solution to a one phase free boundary problem—for an operator with nonstandard growth and non-zero right hand side—and the free boundary is a smooth surface.

2021, 41(6): 2947-2969 doi: 10.3934/dcds.2020392 +[Abstract](503) +[HTML](161) +[PDF](390.66KB)
Abstract:

We prove the existence of a bounded positive solution for the following stationary Schrödinger equation

where \begin{document}$V$\end{document} is a vanishing potential and \begin{document}$f$\end{document} has a sublinear growth at the origin (for example if \begin{document}$f(x,u)$\end{document} is a concave function near the origen). For this purpose we use a Brezis-Kamin argument included in [6]. In addition, if \begin{document}$f$\end{document} has a superlinear growth at infinity, besides the first solution, we obtain a second solution. For this we introduce an auxiliar equation which is variational, however new difficulties appear when handling the compactness. For instance, our approach can be applied for nonlinearities of the type \begin{document}$\rho(x)f(u)$\end{document} where \begin{document}$f$\end{document} is a concave-convex function and \begin{document}$\rho$\end{document} satisfies the \begin{document}$\mathrm{(H)}$\end{document} property introduced in [6]. We also note that we do not impose any integrability assumptions on the function \begin{document}$\rho$\end{document}, which is imposed in most works.

2021, 41(6): 2971-2992 doi: 10.3934/dcds.2020393 +[Abstract](431) +[HTML](163) +[PDF](440.58KB)
Abstract:

We study well-posedness of the complex-valued modified KdV equation (mKdV) on the real line. In particular, we prove local well-posedness of mKdV in modulation spaces \begin{document}$M^{2,p}_{s}( \mathbb{R})$\end{document} for \begin{document}$s \ge \frac14$\end{document} and \begin{document}$2\leq p < \infty$\end{document}. For \begin{document}$s < \frac 14$\end{document}, we show that the solution map for mKdV is not locally uniformly continuous in \begin{document}$M^{2,p}_{s}( \mathbb{R})$\end{document}. By combining this local well-posedness with our previous work (2018) on an a priori global-in-time bound for mKdV in modulation spaces, we also establish global well-posedness of mKdV in \begin{document}$M^{2,p}_{s}( \mathbb{R})$\end{document} for \begin{document}$s \ge \frac14$\end{document} and \begin{document}$2\leq p < \infty$\end{document}.

2021, 41(6): 2993-3020 doi: 10.3934/dcds.2020394 +[Abstract](507) +[HTML](172) +[PDF](450.81KB)
Abstract:

The aim of this paper is to study (regional) fractional Poincaré type inequalities on unbounded domains satisfying the finite ball condition. Both existence and non existence type results on regional fractional inequality are established depending on various conditions on domains and on the range of \begin{document}$s \in (0,1)$\end{document}. The best constant in both regional fractional and fractional Poincaré inequality is characterized for strip like domains \begin{document}$(\omega \times \mathbb{R}^{n-1})$\end{document}, and the results obtained in this direction are analogous to those of the local case. This settles one of the natural questions raised by K. Yeressian in [Asymptotic behavior of elliptic nonlocal equations set in cylinders, Asymptot. Anal. 89, (2014), no 1-2].

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