Communications in Contemporary Mathematics

A classification of local asymptotic profiles and strong unique continuation properties are established for a class of fractional heat equations with a Hardy-type potential, via an Almgren–Poon monotonicity formula combined with a blow-up analysis.

In this paper, the sixth-order Boussinesq equation is studied. We extend the local well-posedness theory for this equation with quadratic and cubic nonlinearities to the high dimensional case. In spite of having the “bad” fourth term $\mathrm{\Delta }u$ in the equation, we derive some dispersive estimates leading to the existence of local solutions which also improves the previous results in the cubic case. In addition, we show persistence of spatial analyticity of solutions for the cubic nonlinearity.

We study the double Yangian associated with the Lie superalgebra ${𝔤𝔩}_{m|n}$. Our main focus is on establishing the Poincaré–Birkhoff–Witt Theorem for the double Yangian and constructing its central elements in the form of coefficients of the quantum contraction. Next, as an application, we introduce reflection algebras, certain left coideal subalgebras of the level 0 double Yangian, and find their presentations by generators and relations.

This paper concerns the structural stability of subsonic flows with a contact discontinuity in a finitely long axisymmetric cylinder. We establish the existence and uniqueness of axisymmetric subsonic flows with a contact discontinuity by prescribing the horizontal mass flux distribution, the swirl velocity, the entropy and the Bernoulli’s quantity at the entrance and the radial velocity at the exit. It can be formulated as a free boundary problem with the contact discontinuity to be determined simultaneously with the flows. Compared with the two-dimensional case, a new difficulty arises due to the singularity near the axis. An invertible modified Lagrangian transformation is introduced to overcome this difficulty and straighten the contact discontinuity. The key elements in our analysis are to utilize the deformation-curl decomposition introduced in [S. Weng and Z. Xin, A deformation-curl decomposition for three dimensional steady Euler equations, Sci. Sin. Math. 49 (2019) 307–320 (in Chinese): doi:10.1360/N012018-00125] to effectively decouple the hyperbolic and elliptic modes in the steady axisymmetric Euler system and to use the implicit function theorem to locate the contact discontinuity.

In this paper, we consider the heat semigroup ${\{W_t\}}_{t>0}$ defined by the combinatorial Laplacian and two subordinated families of ${\{W_t\}}_{t>0}$ on homogeneous trees $X$. We characterize the weights $u$ on $X$ for which the pointwise convergence to initial data of the above families holds for every $f\in L^p(X,\mu ,u)$ with $1\le p<\infty$, where $\mu$ represents the counting measure in $X$. We prove that this convergence property in $X$ is equivalent to the fact that the maximal operator on $t\in (0,R)$, for some $R>0$, defined by the semigroup is bounded from $L^p(X,\mu ,u)$ into $L^p(X,\mu ,v)$ for some weight $v$ on $X$.

It is known that hyperbolic nonautonomous linear delay differential equations in a finite dimensional space are Hyers–Ulam stable and hence shadowable. The converse result is available only in the special case of autonomous and periodic linear delay differential equations with a simple spectrum. In this paper, we prove the converse and hence the equivalence of all three notions in the title for a general class of nonautonomous linear delay differential equations with uniformly bounded coefficients. The importance of the boundedness assumption is shown by an example.

In this paper, we study the asymptotic behavior of sequences of stationary weak solutions to the following Liouville-type equation: