We prove the existence of solutions to the Kuramoto–Sivashinsky equation with low regularity data in function spaces based on the Wiener algebra and in pseudomeasure spaces. In any spatial dimension, we allow the data to have its antiderivative in the Wiener algebra. In one spatial dimension, we also allow data that are in a pseudomeasure space of negative order. In two spatial dimensions, we also allow data that are in a pseudomeasure space one derivative more regular than in the one-dimensional case. In the course of carrying out the existence arguments, we show a parabolic gain of regularity of the solutions as compared to the data. Subsequently, we show that the solutions are in fact analytic at any positive time in the interval of existence.

We identify a class of smooth Banach *-algebras that are differential subalgebras of commutative C*-algebras whose openness of multiplication is completely determined by the topological stable rank of the target C*-algebra. We then show that group algebras of Abelian groups of unbounded exponent fail to have uniformly open convolution. Finally, we completely characterize in the complex case (uniform) openness of multiplication in algebras of continuous functions in terms of the covering dimension.

In a two-dimensional plane, entire solutions of the Allen–Cahn type equation with a finite Morse index necessarily have finite ends. In the case that the nonlinearity is a sine function, all the finite-end solutions have been classified. However, for the classical Allen–Cahn nonlinearity, the structure of the moduli space of these solutions remains unknown. We construct in this paper new finite-end solutions to the Allen–Cahn equation, which will be called fence of saddle solutions, by gluing saddle solutions together.Our construction can be generalized to the case of gluing multiple four-end solutions, with some of their ends being almost parallel.

We study the real-valued modified KdV equation on the real line and the circle in both the focusing and the defocusing cases. By employing the method of commuting flows introduced by Killip and Vişan (2019), we prove global well-posedness in Hs for $0\leq s \lt \tfrac{1}{2}$. On the line, we show how the arguments in the recent article by Harrop-Griffiths, Killip, and Vişan (2020) may be simplified in the higher regularity regime $s\geq 0$. On the circle, we provide an alternative proof of the sharp global well-posedness in L2 due to Kappeler and Topalov (2005) and also extend this to the large-data focusing case.

Consider a flow in $\mathbb{R}^3$ and let K be the biggest invariant subset of some compact region of interest $N \subseteq \mathbb{R}^3$. The set K is often not computable, but the way the flow crosses the boundary of N can provide indirect information about it. For example, classical tools such as Ważewski’s principle or the Poincaré–Hopf theorem can be used to detect whether K is non-empty or contains rest points, respectively. We present a criterion that can establish whether K has a non-trivial homology by looking at the subset of the boundary of N along which the flow is tangent to N. We prove that the criterion is as sharp as possible with the information it uses as an input. We also show that it is algorithmically checkable.

We introduce the notion of the equivariant covering type of a space X on which a finite group G acts and study its properties. The equivariant covering type measures the size of G-equivariant good covers of X and is thus an extension of the covering type of a space, introduced by Karoubi and Weibel. We show that the equivariant covering type is a G-homotopy invariant and describe its relation with other G-invariants, like the equivariant LS-category, G-genus, and the multiplicative structures of equivariant cohomology theories. We also compute the G-covering type of regular G-graphs, give estimates for orientation-preserving actions on surfaces and for the projectivizations of complex representations of G and cohomology spheres. As an application, we derive estimates of sizes of minimal G-triangulations for various G-spaces.

The aim of this article is to study the asymptotic behaviour of non-autonomous stochastic lattice systems. We first show the existence and uniqueness of a pullback measure attractor. Moreover, when deterministic external forcing terms are periodic in time, we show the pullback measure attractors are periodic. We then study the upper semicontinuity of pullback measure attractors as the noise intensity goes to zero. Pullback asymptotic compact for a family of probability measures with respect to probability distributions of the solutions is demonstrated by using uniform a priori estimates for far-field values of solutions.

We present a one-parameter family Fλ of transcendental entire functions with zeros, whose Newton’s method yields wandering domains, coexisting with the basins of the roots of Fλ. Wandering domains for Newton maps of zero-free functions have been built before by, e.g. Buff and Rückert [23] based on the lifting method. This procedure is suited to our Newton maps as members of the class of projectable functions (or maps of the cylinder), i.e. transcendental meromorphic functions f(z) in the complex plane that are semiconjugate, via the exponential, to some map g(w), which may have at most a countable number of essential singularities. In this paper, we make a systematic study of the general relation (dynamical and otherwise) between f and g, and inspect the extension of the logarithmic lifting method of periodic Fatou components to our context, especially for those g of finite-type. We apply these results to characterize the entire functions with zeros whose Newton’s method projects to some map g which is defined at both 0 and $\infty$. The family Fλ is the simplest in this class, and its parameter space shows open sets of λ-values in which the Newton map exhibits wandering or Baker domains, in both cases regions of initial conditions where Newton’s root-findingmethod fails.

In the present paper, we consider the following fractional double phase problem with nonlocal reaction:

\begin{align*}\qquad \quad \left\{\begin{array}{lll}(- \Delta)_p^s u + (- \Delta)_q ^s u + V(\varepsilon x) (|u|^{p - 2} u + |u|^{q - 2} u) = \left(\frac{1}{|x|^\mu} * F(u)\right) f(u) & \ \text{in}\ \mathbb{R}^N, \\u \in W^{s,p}(\mathbb{R}^N) \cap W^{s,q}(\mathbb{R}^N) ,\ u \gt 0 & \ \text{in}\ \mathbb{R}^N,\end{array}\right\}\end{align*}

where ɛ is apositive parameter, $0 \lt s \lt 1$, $2 \leqslant p \lt q \lt \min\{2p, N / s\}$, $0 \lt \mu \lt sp$, $(- \Delta)_t^s$ $(t \in \left\{p,q\right\})$ is the fractional t-Laplace operator, the reaction term $f : \mathbb{R} \mapsto \mathbb{R}$ is continuous, and the potential $V \in C (\mathbb{R}^N , \mathbb{R})$ satisfying a local condition. Using a variational approach and topological tools (the non-standard C1-Nehari manifold analysis and the abstract category theory), multiplicity of positive solutions and concentration properties for the above problem are established. Our results extend and complement some previous contributions related to double phase variational integrals.

In this article, we consider some critical Brézis-Nirenberg problems in dimension $N \geq 3$ that do not have a solution. We prove that a supercritical perturbation can lead to the existence of a positive solution. More precisely, we consider the equation:

\begin{equation*}\left\{\begin{array}{rllll}-\Delta u & = & \lambda u^{q-1} + u^{2^*+ r^\alpha -1} & \mbox{in} & B, \\u & \gt & 0 & \mbox{in} & B, \\u&=&0 & \mbox{on} & \partial B,\\\end{array}\right.\end{equation*}

$B \subset \mathbb{R}^N$

$N\geq 3$

$r=\vert x \vert$

$\alpha \in (0,\min\{N/2,N-2\})$

$q\in [2,2^*]$

$r \in (0,1)$

We study global-in-time dynamics of the stochastic nonlinear beam equations (SNLB) with an additive space-time white noise, posed on the four-dimensional torus. The roughness of the noise leads us to introducing a time-dependent renormalization, after which we show that SNLB is pathwise locally well-posed in all subcritical and most of the critical regimes. For the (renormalized) defocusing cubic SNLB, we establish pathwise global well-posedness below the energy space, by adapting a hybrid argument of Gubinelli-Koch-Oh-Tolomeo (2022) that combines the I-method with a Gronwall-type argument. Lastly, we show almost sure global well-posedness and invariance of the Gibbs measure for the stochastic damped nonlinear beam equations in the defocusing case.

We prove interior boundedness and Hölder continuity for the weak solutions of nonlocal double phase equations in the Heisenberg group $\mathbb{H}^n$. This solves a problem raised by Palatucci and Piccinini et al. in 2022 and 2023 for the nonlinear integro-differential problems in Heisenberg setting. Our proof of the a priori estimates bases on De Giorgi–Nash–Moser theory, where the important ingredients are Caccioppoli-type inequality and Logarithmic estimate. To achieve this goal, we establish a new and crucial Sobolev–Poincaré type inequality in local domain, which may be of independent interest and potential applications.

In this work, the Riemann–Hilbert (RH) problem is employed to study the multiple high-order pole solutions of the cubic Camassa–Holm (cCH) equation with the term characterizing the effect of linear dispersion under zero boundary conditions and nonzero boundary conditions. Under the reflectionless situation, we generalize the residue theorem and obtain the multiple high-order pole solutions of cCH equation by solving an algebraic system. During the process of establishing the solution of RH problem, to simplify the calculations involving the implicitly expressed of variables (x, t) in the solution, we introduce a new scale (y, t) to ensure the solution of RH problem is explicitly expressed with respect to it. Finally, the exact solutions are obtained for cases involving one high-order pole and N high-order poles.

To date, the bestmethodsfor estimating the growth of mean values of arithmetic functions rely on the Voronoï summation formula. By noticing a general pattern in the proof of his summation formula, Voronoï postulated that analogous summation formulas for $\sum a(n)f(n)$ can be obtained with ‘nice’ test functions f(n), provided a(n) is an ‘arithmetic function’. These arithmetic functions a(n) are called so because they are expected to appear as coefficients of some L-functions satisfying certain properties. It has been well-known that the functional equation for a general L-function can be used to derive a Voronoï-type summation identity for that L-function. In this article, we show that such a Voronoï-typesummation identity in fact endows the L-function with some structural properties, yielding in particular the functional equation. We do this by considering Dirichlet series satisfying functional equations involving multiple Gamma factors and show that a given arithmetic function appears as a coefficient of such a Dirichlet series if and only if it satisfies the aforementioned summation formulas.

The article studies an initial boundary valueproblem (ibvp) for the radial solutions of the nonlinear Schrödinger (NLS) equation in a radially symmetric region $\Omega\in \mathbb R^n$ with boundaries. All such regions can be classified into three types: a ball Ω0 centred at origin, a region Ω1 outside a ball, and an n-dimensional annulus Ω2. To study the well-posedness of those ibvps, the function spaces for the boundary data must be specified in terms of the solutions in appropriate Sobolev spaces. It is shown that when $\Omega = \Omega_1$, the ibvp for the NLS equation is locally well-posed in $ C( [0, T^*]; H^s(\Omega_1))$ if the initial data is in $H^s(\Omega_1)$ and boundary data is in $ H^{\frac{2s+1}{4}}(0, T)$ with $s \geq 0$. This is the optimal regularity for the boundary data and cannot be improved. When $\Omega = \Omega_2$, the ibvp is locally well-posed in $ C( [0, T^*]; H^s(\Omega_2))$ if the initial data is in $ H^s(\Omega_2)$ and boundary data is in $ H^{\frac{s+1}{2}}(0, T)$ with $s \geq 0$. In this case, the boundary data requires $1/4$ more derivative compared to the case when $\Omega = \Omega_1$. When $\Omega = \Omega_0$ with n = 2 (the case with n > 2 can be discussed similarly), the ibvp is locally well-posed in $ C( [0, T^*]; H^s(\Omega_0))$ if the initial data is in $ H^s(\Omega_0)$ and boundary data is in $ H^{\frac{s+1}{2}}(0, T)$ with s > 1 (or $s \gt n/2$). Due to the lack of Strichartz estimates for the corresponding boundary integral operator with $ 0 \leq s \leq 1$, the local well-posedness can only be achieved for s > 1. It is noted that the well-posedness results on Ω0 and Ω2 are the first ones for the ibvp of NLS equations in bounded regions of higher dimension.

In this paper, we investigate Kolmogorov-type theorems for small perturbations of degenerate Hamiltonian systems. These systems are index by a parameter ξ as $ H(y,x,\xi) = \langle\omega(\xi),y\rangle {+ \bar h(y,\xi)}+\varepsilon P(y,x,\xi,\varepsilon) $, where ɛ > 0. We assume that the frequency mapping $\omega(\cdot)$, $\bar h(y,\cdot)=O(|y|^2)$ and the perturbation $\varepsilon P(y,x,\cdot, \varepsilon)$ maintain Hölder continuity about ξ. We prove that the persistent invariant tori retain the same frequency as those of the unperturbed tori, under a certain topological degree condition and a weak convexity condition for the frequency mapping. Notably, this paper presents, to our understanding, pioneering results on the KAM theorem under such conditions with only assumption of Hölder continuous dependence of frequency mapping ω on the parameter.

Well-posedness in time-weighted spaces of certain quasilinear (and semilinear) parabolic evolution equations $u'=A(u)u+f(u)$ is established. The focus lies on the case of strict inclusions $\mathrm{dom}(f)\subsetneq \mathrm{dom}(A)$ of the domains of the nonlinearities $u\mapsto f(u)$ and $u\mapsto A(u)$. Based on regularizing effects of parabolic equations it is shown that a semiflow is generated in intermediate spaces. In applications this allows one to derive global existence from weaker a priori estimates. The result is illustrated by examples of chemotaxis systems.

For commuting contractions $T_1,\dots,T_n$ acting on a Hilbert space $\mathscr{H}$ with $T=\prod_{i=1}^n T_i$, we find a necessary and sufficient condition such that $(T_1,\dots,T_n)$ dilates to a commuting tuple of isometries $(V_1,\dots,V_n)$ on the minimal isometric dilation space of T with $V=\prod_{i=1}^nV_i$ being the minimal isometric dilation of T. This isometric dilation provides a commutant lifting of $(T_1, \dots, T_n)$ on the minimal isometric dilation space of T. We construct both Schäffer and Sz. Nagy–Foias-type isometric dilations for $(T_1,\dots,T_n)$ on the minimal dilation spaces of T. Also, a different dilation is constructed when the product T is a $C._0$ contraction, that is, ${T^*}^n \rightarrow 0$ as $n \rightarrow \infty$. As a consequence of these dilation theorems, we obtain different functional models for $(T_1,\dots,T_n)$ in terms of multiplication operators on vectorial Hardy spaces. One notable fact about our models is that the multipliers are all analytic functions in one variable. The dilation when T is a $C._0$ contraction leads to a conditional factorization of T. Several examples have been constructed.

In this paper, we define and study an equivariant analogue of Cohen, Farber and Weinberger’s parametrized topological complexity. We show that several results in the non-equivariant case can be extended to the equivariant case. For example, we establish the fibrewise equivariant homotopy invariance of the sequential equivariant parametrized topological complexity. We obtain several bounds on sequential equivariant topological complexity involving the equivariant category. We also obtain the cohomological lower bound and the dimension-connectivity upper bound on the sequential equivariant parametrized topological complexity. In the end, we use these results to compute the sequential equivariant parametrized topological complexity of equivariant Fadell–Neuwirth fibrations and some equivariant fibrations involving generalized projective product spaces.

We study the geometry of tropical extensions of hyperfields, including the ordinary, signed, and complex tropical hyperfields. We introduce the framework of ‘enriched valuations’ as hyperfield homomorphisms to tropical extensions and show that a notable family of them are relatively algebraically closed. Our main results are hyperfield analogues of Kapranov’s theorem and the Fundamental theorem of tropical geometry. Utilizing these theorems, we introduce fine tropical varieties and prove a structure theorem for them in terms of their initial ideals.