数学物理学报(英文版)

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ON SOME RESULTS RELATED TO SOBER SPACES^∗

Qingguo Li, Mengjie Jin, Hualin Miao, Siheng Chen

数学物理学报(英文版). 2023 (4): 1477-1490. DOI: 10.1007/s10473-023-0401-3

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This paper investigates sober spaces and their related structures from different perspectives. First, we extend the descriptive set theory of second countable sober spaces to first countable sober spaces. We prove that a first countable $T_{0}$ space is sober if and only if it does not contain a $\mathbf{\Pi}_{2}^{0}$-subspace homeomorphic either to $S_{D}$, the natural number set equipped with the Scott topology, or to $S_{1}$, the natural number set equipped with the co-finite topology, and it does not contain any directed closed subset without maximal elements either. Second, we show that if $Y$ is sober, the function space $TOP(X,Y)$ equipped with the Isbell topology (respectively, Scott topology) may be a non-sober space. Furthermore, we provide a uniform construction to $d$-spaces and well-filtered spaces via irreducible subset systems introduced in [9]; we called this an $\mathrm{H}$-well-filtered space. We obtain that, for a $T_{0}$ space $X$ and an $\mathrm{H}$-well-filtered space $Y$, the function space $TOP(X,Y)$ equipped with the Isbell topology is $\mathrm{H}$-well-filtered. Going beyond the aforementioned work, we solve several open problems concerning strong $d$-spaces posed by Xu and Zhao in [11].

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DISTORTION THEOREMS FOR CLASSES OF g-PARAMETRIC STARLIKE MAPPINGS OF REAL ORDER IN $\mathbb{C}^n*$

Hongyan Liu, Zhenhan Tu, Liangpeng XIONG

数学物理学报(英文版). 2023 (4): 1491-1502. DOI: 10.1007/s10473-023-0402-2

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In this paper, we define the class $\widehat{\mathcal {S}}^{\gamma}_g(\mathbb{B}_{\mathbb{X}})$ of $g$-parametric starlike mappings of real order $\gamma$ on the unit ball $\mathbb{B}_{\mathbb{X}}$ in a complex Banach space $\mathbb{X}$, where $g$ is analytic and satisfies certain conditions. By establishing the distortion theorem of the Fr\'{e}chet-derivative type of $\widehat{\mathcal {S}}^{\gamma}_g(\mathbb{B}_{\mathbb{X}})$ with a weak restrictive condition, we further obtain the distortion results of the Jacobi-determinant type and the Fr\'{e}chet-derivative type for the corresponding classes (compared with $\widehat{\mathcal {S}}^{\gamma}_g(\mathbb{B}_{\mathbb{X}})$) defined on the unit polydisc (resp. unit ball with the arbitrary norm) in the space of $n$-dimensional complex variables, $n\geqslant2$. Our results extend the classic distortion theorem of holomorphic functions from the case in one-dimensional complex space to the case in the higher dimensional complex space. The main theorems also generalize and improve some recent works.

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ON THE FIGIEL TYPE PROBLEM AND EXTENSION OF $\varepsilon$-ISOMETRY BETWEEN UNIT SPHERES^∗

Rui LIU, Jifu YIN

数学物理学报(英文版). 2023 (4): 1503-1517. DOI: 10.1007/s10473-023-0403-1

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This paper studies two isometric problems between unit spheres of Banach spaces. In the first part, we introduce and study the Figiel type problem of isometric embeddings between unit spheres. However, the classical Figiel theorem on the whole space cannot be trivially generalized to this case, and this is pointed out by a counterexample. After establishing this, we find a natural necessary condition required by the existence of the Figiel operator. Furthermore, we prove that when $X$ is a space with the T-property, this condition is also sufficient for an isometric embedding $T: S_X\rightarrow S_Y$ to admit the Figiel operator. This answers the Figiel type problem on unit spheres for a large class of spaces. In the second part, we consider the extension of bijective $\varepsilon$-isometries between unit spheres of two Banach spaces. It is shown that every bijective $\varepsilon$-isometry between unit spheres of a local GL-space and another Banach space can be extended to be a bijective $5\varepsilon$-isometry between the corresponding unit balls. In particular, when $\varepsilon=0$, this recovers the MUP for local GL-spaces obtained in [40].

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BANACH SPACES AND INEQUALITIES ASSOCIATED WITH NEW GENERALIZATION OF CESÀRO MATRIX^∗

Feyzi BA͒AR, Hadi ROOPAEI

数学物理学报(英文版). 2023 (4): 1518-1536. DOI: 10.1007/s10473-023-0404-0

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Let the triangle matrix $A^{ru}$ be a generalization of the Cesàro matrix and $U\in\{c_{0},c,\ell_{\infty}\}$. In this study, we essentially deal with the space $U(A^{ru})$ defined by the domain of $A^{ru}$ in the space $U$ and give the bases, and determine the Köthe-Toeplitz, generalized Köthe-Toeplitz and bounded-duals of the space $U(A^{ru})$. We characterize the classes $(\ell_{\infty}(A^{ru}):\ell_{\infty})$, $(\ell_{\infty}(A^{ru}):c)$, $(c(A^{ru}):c)$, and $(U:V(A^{ru}))$ of infinite matrices, where $V$ denotes any given sequence space. Additionally, we also present a Steinhaus type theorem. As an another result of this study, we investigate the $\ell_p$-norm of the matrix $A^{ru}$ and as a result obtaining a generalized version of Hardy's inequality, and some inclusion relations. Moreover, we compute the norm of well-known operators on the matrix domain $\ell_p(A^{ru})$.

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COMPLEX SYMMETRY OF TOEPLITZ OPERATORS OVER THE BIDISK^∗

Maofa Wang, Qi Wu, Kaikai Han

数学物理学报(英文版). 2023 (4): 1537-1546. DOI: 10.1007/s10473-023-0405-z

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In this paper, we investigate the complex symmetric structure of Toeplitz operators $T_\phi$ on the Hardy space over the bidisk. We first characterize the weighted composition operators, $W_{u,v}$ which are $\mathcal{J}$-symmetric and unitary. As a consequence, we characterize conjugations of the form $A_{u,v}$. In addition, a class of conjugations of the form $C_{\lambda, a}$ is introduced. We show that the class of conjugations $C_{\lambda, a}$ coincides with the class of conjugations $A_{u,v}$; we then characterize the complex symmetry of the Toeplitz operators $T_\phi$ with respect to the conjugation $C_{\lambda, a}$.

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NATURALLY REDUCTIVE (α₁, α₂) METRICS^∗

Ju TAN, Ming XU

数学物理学报(英文版). 2023 (4): 1547-1560. DOI: 10.1007/s10473-023-0406-y

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Letting $F$ be a homogeneous $(\alpha_1,\alpha_2)$ metric on the reductive homogeneous manifold $G/H$, we first characterize the natural reductiveness of $F$ as a local $f$-product between naturally reductive Riemannian metrics. Second, we prove the equivalence among several properties of $F$ for its mean Berwald curvature and S-curvature. Finally, we find an explicit flag curvature formula for $G/H$ when $F$ is naturally reductive.

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COMPARISON OF HOMOLOGIES AND AUTOMATIC EXTENSIONS OF INVARIANT DISTRIBUTIONS^∗

Yangyang CHEN

数学物理学报(英文版). 2023 (4): 1561-1570. DOI: 10.1007/s10473-023-0407-x

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Let G be a reductive Nash group, acting on a Nash manifold X. Let Z be a G -stable closed Nash submanifold of X and denote by $U$ the complement of Z in X. Let $\chi$ be a character of G and denote by g the complexified Lie algebra of G. We give a sufficient condition for the natural linear map $H_{k}(g, S(U)\otimes\chi)\rightarrow H_{k}(g, S(X)\otimes\chi)$ between the Lie algebra homologies of Schwartz functions to be an isomorphism. For k=0, by considering the dual, we obtain the automatic extensions of $g$-invariant (twisted by -$\chi$) Schwartz distributions.

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THE NONLINEAR BOUNDARY VALUE PROBLEM FOR k HOLOMORPHIC FUNCTIONS IN $\mathbb{C}^2$^∗

Yanyan CUI, Zunfeng LI, Yonghong XIE, Yuying QIAO

数学物理学报(英文版). 2023 (4): 1571-1586. DOI: 10.1007/s10473-023-0408-9

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k holomorphic functions are a type of generation of holomorphic functions. In this paper, a nonlinear boundary value problem for k holomorphic functions is primarily discussed on generalized polycylinders in $\mathbb{C}^2$. The existence of the solution for the problem is studied in detail with the help of the boundary properties of Cauchy type singular integral operators with a k holomorphic kernel. Furthermore, the integral representation for the solution is obtained.

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BOUNDEDNESS AND COMPACTNESS FOR THE COMMUTATOR OF THE ω-TYPE CALDERÓN-ZYGMUND OPERATOR ON LORENTZ SPACE^∗

Xiangxing TAO, Yuan ZENG, Xiao YU

数学物理学报(英文版). 2023 (4): 1587-1602. DOI: 10.1007/s10473-023-0409-8

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In this paper, the authors consider the $\omega$-type Calderón-Zygmund operator $T_{\omega}$ and the commutator $[b,T_{\omega}]$ generated by a symbol function $b$ on the Lorentz space $L^{p,r}(X)$ over the homogeneous space $(X,d,\mu)$. The boundedness and the compactness of the commutator $[b,T_{\omega}]$ on Lorentz space $L^{p,r}(X)$ are founded for any $p\in (1, \infty)$ and $r\in [1, \infty)$.

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THE WELL-POSEDNESS OF FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS IN COMPLEX BANACH SPACES^∗

Shangquan BU, Gang CAI

数学物理学报(英文版). 2023 (4): 1603-1617. DOI: 10.1007/s10473-023-0410-2

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Let $X$ be a complex Banach space and let $B$ and $C$ be two closed linear operators on $X$ satisfying the condition $D(B)\subset D(C)$, and let $d\in L^1(\mathbb{R}_+)$ and $0 \leq \beta < \alpha\leq 2$. We characterize the well-posedness of the fractional integro-differential equations $D^\alpha u(t) + CD^\beta u(t)$ $= Bu(t) + \int_{-\infty}^t d(t-s)Bu(s){\rm d}s + f(t),\ (0\leq t\leq 2\pi)$ on periodic Lebesgue-Bochner spaces $L^p(\mathbb{T}; X)$ and periodic Besov spaces $B_{p,q}^s(\mathbb{T}; X)$.

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BOUNDEDNESS OF THE CALDERÓN COMMUTATOR WITH A ROUGH KERNEL ON TRIEBEL-LIZORKIN SPACES^∗

Guoen HU, Jie LIU

数学物理学报(英文版). 2023 (4): 1618-1632. DOI: 10.1007/s10473-023-0411-1

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In this paper, we consider the boundedness on Triebel-Lizorkin spaces for the $d$-dimensional Calderón commutator defined by $T_{\Omega,a}f(x)={\rm p.\,v.}\int_{\mathbb{R}^d}\frac{\Omega(x-y)}{|x-y|^{d+1}}\big(a(x)-a(y)\big)f(y){\rm d}y,$ where $\Omega$ is homogeneous of degree zero, integrable on $S^{d-1}$ and has a vanishing moment of order one, and $a$ is a function on $\mathbb{R}^d$ such that $\nabla a\in L^{\infty}(\mathbb{R}^d)$. We prove that if $1<p,\,q<\infty$ and $\Omega\in L(\log L)^{2\tilde{q}}(S^{d-1})$ with $\tilde{q}=\max\{1/q,\,1/q'\}$, then $T_{\Omega,a}$ is bounded on Triebel-Lizorkin spaces $\dot{F}_{p}^{0,q}(\mathbb{R}^d)$.

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LOCAL BIFURCATION OF STEADY ALMOST PERIODIC WATER WAVES WITH CONSTANT VORTICITY^∗

Wei LUO, Zhaoyang YIN

数学物理学报(英文版). 2023 (4): 1633-1644. DOI: 10.1007/s10473-023-0412-0

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In this paper we investigate the traveling wave solution of the two dimensional Euler equations with gravity at the free surface over a flat bed. We assume that the free surface is almost periodic in the horizontal direction. Using conformal mappings, one can change the free boundary problem into a fixed boundary problem for some unknown functions with the boundary condition. By virtue of the Hilbert transform, the problem is equivalent to a quasilinear pseudodifferential equation for an almost periodic function of one variable. The bifurcation theory ensures that we can obtain an existence result. Our existence result generalizes and covers the recent result in [15]. Moreover, our result implies a non-uniqueness result at the same bifurcation point.

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TIME PERIODIC SOLUTIONS TO THE EVOLUTIONARY OSEEN MODEL FOR A GENERALIZED NEWTONIAN INCOMPRESSIBLE FLUID^∗

Jinxia CEN, Stanis law MIGÓRSKI, Emilio VILCHES, Shengda ZENG

数学物理学报(英文版). 2023 (4): 1645-1667. DOI: 10.1007/s10473-023-0413-z

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In this paper we study a nonstationary Oseen model for a generalized Newtonian incompressible fluid with a time periodic condition and a multivalued, nonmonotone friction law. First, a variational formulation of the model is obtained; that is a nonlinear boundary hemivariational inequality of parabolic type for the velocity field. Then, an abstract first-order evolutionary hemivariational inequality in the framework of an evolution triple of spaces is investigated. Under mild assumptions, the nonemptiness and weak compactness of the set of periodic solutions to the abstract inequality are proven. Furthermore, a uniqueness theorem for the abstract inequality is established by using a monotonicity argument. Finally, we employ the theoretical results to examine the nonstationary Oseen model.

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THE GLOBAL EXISTENCE OF BV SOLUTIONS OF THE ISENTROPIC p-SYSTEM WITH LARGE INITIAL DATA^∗

Fei WU, Zejun WANG, Fangqi CHEN

数学物理学报(英文版). 2023 (4): 1668-1674. DOI: 10.1007/s10473-023-0414-y

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In this paper, we study the global existence of BV solutions of the initial value problem for the isentropic p-system, where the state equation of the gas is given by $P=Av^{-\gamma}$. For $\gamma>1$, the general existence result for large initial data has not been obtained. By using the Glimm scheme, Nishida, Smoller and Diperna successively obtained the global existence results for $(\gamma-1)\mbox{TV}(v_0(x),u_0(x))$ being small. In the present paper, by adopting a rescaling technique, we improve these results and obtain the global existence result under the condition that $(\gamma-1)^{\gamma+1}({\rm TV}(v_{0}(x)))^{\gamma-1}(\mbox{TV}(u_{0}(x)))^{2}$ is small, which implies that, for fixed $\gamma>1$, either $\mbox{TV}(v_{0}(x))$ or $\mbox{TV}(u_{0}(x))$ can be arbitrarily large.

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THE REGULARITY AND UNIQUENESS OF A GLOBAL SOLUTION TO THE ISENTROPIC NAVIER-STOKES EQUATION WITH ROUGH INITIAL DATA^∗

Haitao WANG, Xiongtao ZHANG

数学物理学报(英文版). 2023 (4): 1675-1716. DOI: 10.1007/s10473-023-0415-x

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A global weak solution to the isentropic Navier-Stokes equation with initial data around a constant state in the $L^1\cap$ BV class was constructed in [1]. In the current paper, we will continue to study the uniqueness and regularity of the constructed solution. The key ingredients are the Hölder continuity estimates of the heat kernel in both spatial and time variables. With these finer estimates, we obtain higher order regularity of the constructed solution to Navier-Stokes equation, so that all of the derivatives in the equation of conservative form are in the strong sense. Moreover, this regularity also allows us to identify a function space such that the stability of the solutions can be established there, which eventually implies the uniqueness.

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THE SHOCK WAVES FOR A MIXED-TYPE SYSTEM FROM CHEMOTAXIS^∗

Fen HE, Zhen WANG, Tingting CHEN

数学物理学报(英文版). 2023 (4): 1717-1734. DOI: 10.1007/s10473-023-0416-9

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In this paper, we study the shock waves for a mixed-type system from chemotaxis. We are concerned with the jump conditions for the left state which is located in the elliptical region and the right state in the hyperbolic region. Under the generalized entropy conditions, we find that there are different shock wave structures for different parameters. To guarantee the uniqueness of the solutions, we obtain the admissible shock waves which satisfy the generalized entropy condition in both parameters. Finally, we construct the Riemann solutions in some solvable regions.

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ASYMPTOTIC STABILITY OF SHOCK WAVES FOR THE OUTFLOW PROBLEM OF A HEAT-CONDUCTIVE IDEAL GAS WITHOUT VISCOSITY^∗

Lili FAN, Meichen HOU

数学物理学报(英文版). 2023 (4): 1735-1766. DOI: 10.1007/s10473-023-0417-8

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This paper is concerned with an ideal polytropic model of non-viscous and heat-conductive gas in a one-dimensional half space. We focus our attention on the outflow problem when the flow velocity on the boundary is negative and we prove the stability of the viscous shock wave and its superposition with the boundary layer under some smallness conditions. Our waves occur in the subsonic area. The intrinsic properties of our system are more challenging in mathematical analysis, however, in the subsonic area, the lack of a boundary condition on the density provides us with a special manner for defining the shift for the viscous shock wave, and helps us to construct the asymptotic profiles successfully. New weighted energy estimates are introduced and the perturbations on the boundary are handled by some subtle estimates.

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LARGE DEVIATIONS FOR TOP EIGENVALUES OF ß -JACOBI ENSEMBLES AT SCALING TEMPERATURES^∗

Liangzhen LEI, Yutao MA

数学物理学报(英文版). 2023 (4): 1767-1780. DOI: 10.1007/s10473-023-0418-7

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Let $\lambda=(\lambda_1, \cdots, \lambda_n)$ be $\beta$-Jacobi ensembles with parameters $p_1, p_2, n$ and $\beta,$ with $\beta$ varying with $n.$ Set $\gamma=\lim\limits_{n\rightarrow\infty}\frac{n}{p_1}$ and $\sigma=\lim\limits_{n\rightarrow\infty}\frac{p_1}{p_2}.$ Suppose that $\lim\limits_{n\to\infty}\frac{\log n}{\beta n}=0$ and $0\le \sigma\gamma< 1.$ We offer the large deviation for $\frac{p_1+p_2}{p_1}\max\limits_{1\le i\le n}\lambda_{i}$ when $\gamma>0$ via the large deviation of the corresponding empirical measure and via a direct estimate, respectively, when $\gamma=0.$

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THE EXISTENCE AND CONCENTRATION OF GROUND STATE SIGN-CHANGING SOLUTIONS FOR KIRCHHOFF-TYPE EQUATIONS WITH A STEEP POTENTIAL WELL^∗

Menghui WU, Chunlei TANG

数学物理学报(英文版). 2023 (4): 1781-1799. DOI: 10.1007/s10473-023-0419-6

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In this paper, we consider the nonlinear Kirchhoff type equation with a steep potential well \begin{equation*} -\Big(a+b\int_{\mathbb{R}^{3}}|\nabla u|^{2}\,{\rm d}x\Big)\Delta u+\lambda V(x)u=f(u) \qquad {\rm in}\ \mathbb{R}^{3}, \end{equation*} where $a,b>0$ are constants, $\lambda$ is a positive parameter, $V\in C(\mathbb{R}^{3}, \mathbb{R})$ is a steep potential well and the nonlinearity $f\in C(\mathbb{R}, \mathbb{R})$ satisfies certain assumptions. By applying a sign-changing Nehari manifold combined with the method of constructing a sign-changing $(PS)_{C}$ sequence, we obtain the existence of ground state sign-changing solutions with precisely two nodal domains when $\lambda$ is large enough, and find that its energy is strictly larger than twice that of the ground state solutions. In addition, we also prove the concentration of ground state sign-changing solutions.

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THE ZERO LIMIT OF THERMAL DIFFUSIVITY FOR THE 2D DENSITY-DEPENDENT BOUSSINESQ EQUATIONS^∗

Xia YE, Yanxia XU, Zejia WANG

数学物理学报(英文版). 2023 (4): 1800-1818. DOI: 10.1007/s10473-023-0420-0

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This paper is concerned with the asymptotic behavior of solutions to the initial boundary problem of the two-dimensional density-dependent Boussinesq equations. It is shown that the solutions of the Boussinesq equations converge to those of zero thermal diffusivity Boussinesq equations as the thermal diffusivity tends to zero, and the convergence rate is established. In addition, we prove that the boundary-layer thickness is of the value $\delta(k)=k^{\alpha}$ with any $ \alpha\in(0,1/4)$ for a small diffusivity coefficient $k>0$, and we also find a function to describe the properties of the boundary layer.

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MULTIPLE POSITIVE SOLUTIONS TO A CLASS OF MODIFIED NONLINEAR SCHRÖDINGER EQUATION IN A HIGH DIMENSION^∗

Yansheng ZHONG, Yongqing LI

数学物理学报(英文版). 2023 (4): 1819-1840. DOI: 10.1007/s10473-023-0421-z

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The existence and multiplicity of positive solutions for equation (1.1) with the new critical exponent $4<p<2\cdot 2^*$ shall be investigated in a high dimension. The conclusions extend the relative results recently attained in \cite{1} for the one-dimensional case. More precisely, as the coefficient $a(x)$ in the nonlinearity is sign-changing, the modified term $2(\Delta (|u|^2))u$ is still helpful for obtaining multiple positive solutions in a high dimension, even if a sign condition like $\int_{\mathbb{R}^N}a(x)e_1^p{\rm d}x<0$ (also named “a necessary condition” see [2,3]) does not hold.

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BLOW-UP SOLUTIONS OF TWO-COUPLED NONLINEAR SCHR ÖDINGER EQUATIONS IN THE RADIAL CASE^∗

Qianqian BAI, Xiaoguang LI, Li ZHANG

数学物理学报(英文版). 2023 (4): 1841-1864. DOI: 10.1007/s10473-023-0422-y

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We consider the blow-up solutions to the following coupled nonlinear Schrödinger equations \begin{equation*} \left\{ \begin{aligned} &{\rm i}u_{t}+\Delta u+(|u|^{2p}+\beta|u|^{p-1}|v|^{p+1})u=0,\\ &{\rm i}v_{t}+\Delta v+(|v|^{2p}+\beta|v|^{p-1}|u|^{p+1})v=0,\\ &u(0,x)=u_{0}(x),\ \ \ \ v(0,x)=v_{0}(x),\ \ x\in \mathbb{R}^{N},\ t\geq0. \end{aligned} \right. \end{equation*} On the basis of the conservation of mass and energy, we establish two sufficient conditions to obtain the existence of a blow-up for radially symmetric solutions. These results improve the blow-up result of Li and Wu [10] by dropping the hypothesis of finite variance ($(|x|u_{0},|x|v_{0})\in L^{2}(\mathbb{R}^{N})\times L^{2}(\mathbb{R}^{N})$).

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BLOW-UP SOLUTIONS OF TWO-COUPLED NONLINEAR SCHR ÖDINGER EQUATIONS IN THE RADIAL CASE^∗

Qianqian BAI, Xiaoguang LI, Li ZHANG

数学物理学报(英文版). 2023 (4): 1852-1864. DOI: 10.1007/s10473-023-0423-x

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We consider the blow-up solutions to the following coupled nonlinear Schrödinger equations \begin{equation*} \left\{ \begin{aligned} &{\rm i}u_{t}+\Delta u+(|u|^{2p}+\beta|u|^{p-1}|v|^{p+1})u=0,\\ &{\rm i}v_{t}+\Delta v+(|v|^{2p}+\beta|v|^{p-1}|u|^{p+1})v=0,\\ &u(0,x)=u_{0}(x),\ \ \ \ v(0,x)=v_{0}(x),\ \ x\in \mathbb{R}^{N},\ t\geq0. \end{aligned} \right. \end{equation*} On the basis of the conservation of mass and energy, we establish two sufficient conditions to obtain the existence of a blow-up for radially symmetric solutions. These results improve the blow-up result of Li and Wu [10] by dropping the hypothesis of finite variance ($(|x|u_{0},|x|v_{0})\in L^{2}(\mathbb{R}^{N})\times L^{2}(\mathbb{R}^{N})$).

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ASYMPTOTICAL STABILITY OF NEUTRAL REACTION-DIFFUSION EQUATIONS WITH PCAS AND THEIR FINITE ELEMENT METHODS^∗

Hao HAN, Chengjian ZHANG

数学物理学报(英文版). 2023 (4): 1865-1880. DOI: 10.1007/s10473-023-0424-9

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This paper focuses on the analytical and numerical asymptotical stability of neutral reaction-diffusion equations with piecewise continuous arguments. First, for the analytical solutions of the equations, we derive their expressions and asymptotical stability criteria. Second, for the semi-discrete and one-parameter fully-discrete finite element methods solving the above equations, we work out the sufficient conditions for assuring that the finite element solutions are asymptotically stable. Finally, with a typical example with numerical experiments, we illustrate the applicability of the obtained theoretical results.

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THE GLOBAL SOLUTION AND BLOWUP OF A SPATIOTEMPORAL EIT PROBLEM WITH A DYNAMICAL BOUNDARY CONDITION^∗

Minghong XIE, Zhong TAN

数学物理学报(英文版). 2023 (4): 1881-1914. DOI: 10.1007/s10473-023-0425-8

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We study a spatiotemporal EIT problem with a dynamical boundary condition for the fractional Dirichlet-to-Neumann operator with a critical exponent. There are three major ingredients in this paper. The first is the finite time blowup and the decay estimate of the global solution with a lower-energy initial value. The second ingredient is the $L^q(2\leq q<\infty)$ estimate of the global solution applying the Moser iteration, which allows us to show that any global solution is a classical solution. The third, which is the main ingredient of this paper, explores the long time asymptotic behavior of global solutions close to the stationary solution and the bubbling phenomenons by means of a concentration compactness principle.

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THE VARIATIONAL PRINCIPLE FOR THE PACKING ENTROPY OF NONAUTONOMOUS DYNAMICAL SYSTEMS^∗

Ruifeng ZHANG, Jianghui ZHU

数学物理学报(英文版). 2023 (4): 1915-1924. DOI: 10.1007/s10473-023-0426-7

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Let $(X,\phi)$ be a nonautonomous dynamical system. In this paper, we introduce the notions of packing topological entropy and measure-theoretical upper entropy for nonautonomous dynamical systems. Moreover, we establish the variational principle between the packing topological entropy and the measure-theoretical upper entropy.

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