Acta Mathematica Scientia (Series B)
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ISSN 0252-9602
CN  42-1227/O
25 August 2022, Volume 42 Issue 4 Previous Issue   
Articles
MAXIMAL $L^1$-REGULARITY OF GENERATORS FOR BOUNDED ANALYTIC SEMIGROUPS IN BANACH SPACES
Myong-Hwan RI, Reinhard FARWIG
Acta mathematica scientia,Series B. 2022, 42 (4):  1261-1272.  DOI: 10.1007/s10473-022-0401-8
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In this paper, we prove that the generator of any bounded analytic semigroup in $(\theta,1)$-type real interpolation of its domain and underlying Banach space has maximal $L^1$-regularity, using a duality argument combined with the result of maximal continuous regularity. As an application, we consider maximal $L^1$-regularity of the Dirichlet-Laplacian and the Stokes operator in inhomogeneous $B^s_{q,1}$-type Besov spaces on domains of $\mathbb R^n$, $n\geq 2$.
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$\mathcal{O}(t^{-\beta})$-SYNCHRONIZATION AND ASYMPTOTIC SYNCHRONIZATION OF DELAYED FRACTIONAL ORDER NEURAL NETWORKS
Anbalagan PRATAP, Ramachandran RAJA, Jinde CAO, Chuangxia HUANG, Chuangxia HUANG, Ovidiu BAGDASAR
Acta mathematica scientia,Series B. 2022, 42 (4):  1273-1292.  DOI: 10.1007/s10473-022-0402-7
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This article explores the $\mathcal{O}(t^{-\beta})$ synchronization and asymptotic synchronization for fractional order BAM neural networks (FBAMNNs) with discrete delays, distributed delays and non-identical perturbations. By designing a state feedback control law and a new kind of fractional order Lyapunov functional, a new set of algebraic sufficient conditions are derived to guarantee the $\mathcal{O}(t^{-\beta})$ Synchronization and asymptotic synchronization of the considered FBAMNNs model; this can easily be evaluated without using a MATLAB LMI control toolbox. Finally, two numerical examples, along with the simulation results, illustrate the correctness and viability of the exhibited synchronization results.
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GLOBAL WELL-POSEDNESS OF THE 2D BOUSSINESQ EQUATIONS WITH PARTIAL DISSIPATION
Xueting Jin, Yuelong Xiao, Huan Yu
Acta mathematica scientia,Series B. 2022, 42 (4):  1293-1309.  DOI: 10.1007/s10473-022-0403-6
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In this paper, we prove the global well-posedness of the 2D Boussinesq equations with three kinds of partial dissipation; among these the initial data $(u_0,\theta_0)$ is required such that its own and the derivative of one of its directions $(x,y)$ are assumed to be $L^2(\mathbb R^2)$. Our results only need the lower regularity of the initial data, which ensures the uniqueness of the solutions.
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WEIGHTED NORM INEQUALITIES FOR COMMUTATORS OF THE KATO SQUARE ROOT OF SECOND ORDER ELLIPTIC OPERATORS ON $\mathbb R^n$
Yanping CHEN, Yong DING, Kai ZHU
Acta mathematica scientia,Series B. 2022, 42 (4):  1310-1332.  DOI: 10.1007/s10473-022-0404-5
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Let $L=-\mathrm{div}(A\nabla)$ be a second order divergence form elliptic operator with bounded measurable coefficients in ${\Bbb R}^n$. We establish weighted $L^p$ norm inequalities for commutators generated by $\sqrt{L}$ and Lipschitz functions, where the range of $p$ is different from $(1,\infty)$, and we isolate the right class of weights introduced by Auscher and Martell. In this work, we use good-$\lambda$ inequality with two parameters through the weighted boundedness of Riesz transforms $\nabla L^{-1/2}$. Our result recovers, in some sense, a previous result of Hofmann.
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ITERATIVE ALGORITHMS FOR SYSTEM OF VARIATIONAL INCLUSIONS IN HADAMARD MANIFOLDS
Qamrul Hasan ANSARI, Feeroz BABU, D. R. SAHU
Acta mathematica scientia,Series B. 2022, 42 (4):  1333-1356.  DOI: 10.1007/s10473-022-0405-4
In this paper, we consider system of variational inclusions and its several spacial cases, namely, alternating point problems, system of variational inequalities, etc., in the setting of Hadamard manifolds. We propose an iterative algorithm for solving system of variational inclusions and study its convergence analysis. Several special cases of the proposed algorithm and convergence result are also presented. We present application to constraints minimization problems for bifunctions in the setting of Hadamard manifolds. At the end, we illustrate proposed algorithms and convergence analysis by a numerical example. The algorithms and convergence results of this paper either improve or extend known algorithms and convergence results from linear structure to Hadamard manifolds.
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THE GENERALIZED HYPERSTABILITY OF GENERAL LINEAR EQUATION IN QUASI-2-BANACH SPACE
Ravinder Kumar SHARMA, Sumit CHANDOK
Acta mathematica scientia,Series B. 2022, 42 (4):  1357-1372.  DOI: 10.1007/s10473-022-0406-3
In this paper, we study the hyperstability for the general linear equation \[f(ax+by)=Af(x)+Bf(y) \] in the setting of complete quasi-2-Banach spaces. We first extend the main fixed point result of Brzdȩk and Ciepliński (Acta Mathematica Scientia, 2018, ${\bf 38B}$(2): 377-390) to quasi-2-Banach spaces by defining an equivalent quasi-2-Banach space. Then we use this result to generalize the main results on the hyperstability for the general linear equation in quasi-2-Banach spaces. Our results improve and generalize many results of literature.
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NO-ARBITRAGE SYMMETRIES
Iván DEGANO, Sebastián FERRANDO, Alfredo GONZÁLEZ
Acta mathematica scientia,Series B. 2022, 42 (4):  1373-1402.  DOI: 10.1007/s10473-022-0407-2
The no-arbitrage property is widely accepted to be a centerpiece of modern financial mathematics and could be considered to be a financial law applicable to a large class of (idealized) markets. This paper addresses the following basic question: can one characterize the class of transformations that leave the law of no-arbitrage invariant? We provide a geometric formalization of this question in a non probabilistic setting of discrete time-the so-called trajectorial models. The paper then characterizes, in a local sense, the no-arbitrage symmetries and illustrates their meaning with a detailed example. Our context makes the result available to the stochastic setting as a special case.
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THE DISCRETE ORLICZ-MINKOWSKI PROBLEM FOR $p$-CAPACITY
Lewen JI, Zhihui YANG
Acta mathematica scientia,Series B. 2022, 42 (4):  1403-1413.  DOI: 10.1007/s10473-022-0408-1
In this paper, we demonstrate the existence part of the discrete Orlicz-Minkowski problem for $p$-capacity when 1<p<2.
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THE EXISTENCE AND BLOW-UP OF THE RADIAL SOLUTIONS OF A ${(k_{1},k_{2})}$-HESSIAN SYSTEM INVOLVING A NONLINEAR OPERATOR AND GRADIENT
Guotao WANG, Zedong YANG, Jiafa XU, Lihong ZHANG
Acta mathematica scientia,Series B. 2022, 42 (4):  1414-1426.  DOI: 10.1007/s10473-022-0409-0
In this paper, we are concerned with the existence of the positive bounded and blow-up radial solutions of the $(k_{1},k_{2})$-Hessian system \begin{equation*} \begin{split} \left\{\begin{array}{l}{\mathcal{G} (K_{1}^{\frac{1}{k_{1}}}) K_{1}^{\frac{1}{k_{1}}}=b_{1}(|x|) g_{1}(z_{1}, z_{2}), ~~x \in \mathbb{R}^{N}}, \\ {\mathcal{G}(K_{2}^{\frac{1}{k_{2}}}) K_{2}^{\frac{1}{k_{2}}}=b_{2}(|x|) g_{2}(z_{1}, z_{2}), ~~x \in \mathbb{R}^{N}},\end{array}\right. \end{split} \end{equation*} where $\mathcal{G}$ is a nonlinear operator, $K_{i}=S_{k_{i}}\left(\lambda\left(D^{2} z_{i}\right)\right)+\psi_{i}(|x|)|\nabla z_{i}|^{k_{i}},i=1,2.$ Under the appropriate conditions on $g_{1}$ and $g_{2}$, our main results are obtained by using the monotone iterative method and the Arzela-Ascoli theorem. Furthermore, our main results also extend the previous existence results for both the single equation and systems.
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A NEW PROOF OF GAFFNEY’S INEQUALITY FOR DIFFERENTIAL FORMS ON MANIFOLDS-WITH-BOUNDARY: THE VARIATIONAL APPROACH À LA KOZONO–YANAGISAWA
Siran LI
Acta mathematica scientia,Series B. 2022, 42 (4):  1427-1452.  DOI: 10.1007/s10473-022-0410-7
Let $(M,g_0)$ be a compact Riemannian manifold-with-boundary. We present a new proof of the classical Gaffney inequality for differential forms in boundary value spaces over $M$, via a variational approach $à la$ Kozono-Yanagisawa [$L^r$-variational inequality for vector fields and the Helmholtz-Weyl decomposition in bounded domains, Indiana Univ. Math. J. $\textbf{58}$ (2009), 1853-1920], combined with global computations based on the Bochner technique.
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THE MULTIPLICITY AND CONCENTRATION OF POSITIVE SOLUTIONS FOR THE KIRCHHOFF-CHOQUARD EQUATION WITH MAGNETIC FIELDS
Li WANG, Kun CHENG, Jixiu WANG
Acta mathematica scientia,Series B. 2022, 42 (4):  1453-1484.  DOI: 10.1007/s10473-022-0411-6
In this paper, we study the multiplicity and concentration of positive solutions for the following fractional Kirchhoff-Choquard equation with magnetic fields: \begin{equation*} (a\varepsilon^{2s}+b\varepsilon^{4s-3}[u]^2_{\varepsilon,A/\varepsilon}) (-\Delta)_{A/\varepsilon}^{s} u+V(x)u = \varepsilon^{-\alpha}(I_\alpha*F(|u|^2))f(|u|^2)u\ \ \text{in }\ \mathbb{R}^3. \end{equation*} Here $\varepsilon > 0$ is a small parameter, $a,b > 0$ are constants, $s \in (0% \frac{3} {4} ,1), (-\Delta)_{A}^{s}$ is the fractional magnetic Laplacian, $A: \mathbb{R}^3 \to \mathbb{R}^3$ is a smooth magnetic potential, $I_{\alpha}=\frac{\Gamma(\frac{3-\alpha}{2})}{2^{\alpha}\pi^{\frac{3}{2}}\Gamma(\frac{\alpha}{2})}\cdot\frac{1}{|x|^{\alpha} }$ is the Riesz potential, the potential $V$ is a positive continuous function having a local minimum, and $f: \mathbb{R} \to \mathbb{R}$ is a $C^1$ subcritical nonlinearity. Under some proper assumptions regarding $V$ and $f, $ we show the multiplicity and concentration of positive solutions with the topology of the set $M:= \{x \in \mathbb{R}^3 : V (x) = \inf V \}$ by applying the penalization method and Ljusternik-Schnirelmann theory for the above equation.
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TWO REGULARIZATION METHODS FOR IDENTIFYING THE SOURCE TERM PROBLEM ON THE TIME-FRACTIONAL DIFFUSION EQUATION WITH A HYPER-BESSEL OPERATOR
Fan YANG, Qiaoxi SUN, Xiaoxiao LI
Acta mathematica scientia,Series B. 2022, 42 (4):  1485-1518.  DOI: 10.1007/s10473-022-0412-5
In this paper, we consider the inverse problem for identifying the source term of the time-fractional equation with a hyper-Bessel operator. First, we prove that this inverse problem is ill-posed, and give the conditional stability. Then, we give the optimal error bound for this inverse problem. Next, we use the fractional Tikhonov regularization method and the fractional Landweber iterative regularization method to restore the stability of the ill-posed problem, and give corresponding error estimates under different regularization parameter selection rules. Finally, we verify the effectiveness of the method through numerical examples.
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THE STABILITY OF THE DELTA WAVE TO PRESSURELESS EULER EQUATIONS WITH VISCOUS AND FLUX PERTURBATIONS
Sijie LIU, Wancheng SHENG
Acta mathematica scientia,Series B. 2022, 42 (4):  1519-1535.  DOI: 10.1007/s10473-022-0413-4
This paper is concerned with the pressureless Euler equations with viscous and flux perturbations. The existence of Riemann solutions to the pressureless Euler equations with viscous and flux perturbations is obtained. We show the stability of the delta wave of the pressureless Euler equations to the perturbations; that is, the limit solution of the pressureless Euler equations with viscous and flux perturbations is the delta wave solution of the pressureless Euler equations as the viscous and flux perturbation simultaneously vanish in the case $u_->u_+$.
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TRANSONIC SHOCK SOLUTIONS TO THE EULER SYSTEM IN DIVERGENT-CONVERGENT NOZZLES
Ben DUAN, Ao LAN, Zhen LUO
Acta mathematica scientia,Series B. 2022, 42 (4):  1536-1546.  DOI: 10.1007/s10473-022-0414-3
In this paper, we study the transonic shock solutions to the steady Euler system in a quasi-one-dimensional divergent-convergent nozzle. For a given physical supersonic inflow at the entrance, we obtain exactly two non-isentropic transonic shock solutions for the exit pressure lying in a suitable range. In addition, we establish the monotonicity between the location of the transonic shock and the pressure downstream.
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SPACE-LIKE BLASCHKE ISOPARAMETRIC SUBMANIFOLDS IN THE LIGHT-CONE OF CONSTANT SCALAR CURVATURE
Hongru SONG, Ximin LIU
Acta mathematica scientia,Series B. 2022, 42 (4):  1547-1568.  DOI: 10.1007/s10473-022-0415-2
Let $\mathfrak{C}^{m+p+1}_s\subset\mathbb{R}^{m+p+2}_{s+1}$ ($m\geq 2$, $p\geq 1$, $0\leq s\leq p$) be the standard (punched) light-cone in the Lorentzian space $\mathbb{R}^{m+p+2}_{s+1}$, and let $Y:M^m\to \mathfrak{C}^{m+p+1}_s$ be a space-like immersed submanifold of dimension $m$. Then, in addition to the induced metric $g$ on $M^m$, there are three other important invariants of $Y$: the Blaschke tensor $A$, the conic second fundamental form $B$, and the conic Möbius form $C$; these are naturally defined by $Y$ and are all invariant under the group of rigid motions on $\mathfrak{C}^{m+p+1}_s$. In particular, $g,A,B,C$ form a complete invariant system for $Y$, as was originally shown by C. P. Wang for the case in which $s=0$. The submanifold $Y$ is said to be Blaschke isoparametric if its conic Möbius form $C$ vanishes identically and all of its Blaschke eigenvalues are constant. In this paper, we study the space-like Blaschke isoparametric submanifolds of a general codimension in the light-cone $\mathfrak{C}^{m+p+1}_s$ for the extremal case in which $s=p$. We obtain a complete classification theorem for all the $m$-dimensional space-like Blaschke isoparametric submanifolds in $\mathfrak{C}^{m+p+1}_p$ of constant scalar curvature, and of two distinct Blaschke eigenvalues.
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TWO DIMENSIONAL SUBSONIC AND SUBSONIC-SONIC SPIRAL FLOWS OUTSIDE A POROUS BODY
Shangkun WENG, Zihao ZHANG
Acta mathematica scientia,Series B. 2022, 42 (4):  1569-1584.  DOI: 10.1007/s10473-022-0416-1
In this paper, we investigate two dimensional subsonic and subsonic-sonic spiral flows outside a porous body. The existence and uniqueness of the subsonic spiral flow are obtained via variational formulation, which tends to a given radially symmetric subsonic spiral flow at far field. The optimal decay rate at far field is also derived by Kelvin’s transformation and some elliptic estimates. By extracting spiral subsonic solutions as the approximate sequences, we obtain the spiral subsonic-sonic limit solution by utilizing the compensated compactness. The main ingredients of our analysis are methods of calculus of variations, the theory of second-order quasilinear equations and the compensated compactness framework.
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CLASSICAL SOLUTIONS OF THE 3D COMPRESSIBLE FLUID-PARTICLE SYSTEM WITH A MAGNETIC FIELD
Bingyuan HUANG, Shijin DING, Riqing WU
Acta mathematica scientia,Series B. 2022, 42 (4):  1585-1606.  DOI: 10.1007/s10473-022-0417-0
This paper addresses the 3-D Cauchy problem of a fluid-particle system with a magnetic field. First, the local classical solutions of the linearized model on the sphere $B_{r} $ are obtained by some a priori estimates that do not depend on the radius $r$. Second, the classical solutions of the linearized model in $\mathbb{R}^{3}$ are obtained by combining the continuation and compactness methods. Finally, the classical solutions of the original system are proved by use of the picard iteration argument and the energy method.
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ON THE DIMENSION OF THE DIVERGENCE SET OF THE OSTROVSKY EQUATION
Yajuan ZHAO, Yongsheng LI, Wei YAN, Xiangqian YAN
Acta mathematica scientia,Series B. 2022, 42 (4):  1607-1620.  DOI: 10.1007/s10473-022-0418-z
We investigate the refined Carleson's problem of the free Ostrovsky equation \begin{equation*} \left\{ \begin{aligned} & u_t+\partial_x^3u+\partial_x^{-1}u=0,\\ & u(x,0)=f(x), \end{aligned} \right. \end{equation*} where $(x,t)\in\mathbb{R}\times\mathbb{R}$ and $f\in H^s(\mathbb{R})$. We illustrate the Hausdorff dimension of the divergence set for the Ostrovsky equation \begin{equation*} \alpha_{1,U}(s)=1-2s,\quad \frac{1}{4}\leq s\leq\frac{1}{2}. \end{equation*}
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SOME EQUIVALENT CONDITIONS OF PROXIMINALITY IN NONREFLEXIVE BANACH SPACES
Zihou ZHANG, Yu ZHOU, Chunyan LIU, Jing ZHOU
Acta mathematica scientia,Series B. 2022, 42 (4):  1621-1630.  DOI: 10.1007/s10473-022-0419-y
In this paper, we discuss the relation between $\tau$-strongly Chebyshev, approximatively $\tau$-compact $k$-Chebyshev, approximatively $\tau$-compact, $\tau$-strongly proximinal and proximinal sets, where $\tau$ is the norm or the weak topology. We give some equivalent conditions regarding the above proximinality. Furthermore, we also propose the necessary and sufficient conditions that a half-space is $\tau$-strongly proximinal, approximatively $\tau$-compact and $\tau$-strongly Chebyshev.
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A UNIQUENESS THEOREM FOR HOLOMORPHIC MAPPINGS IN THE DISK SHARING TOTALLY GEODESIC HYPERSURFACES
Jiaxing HUANG, Tuen Wai NG
Acta mathematica scientia,Series B. 2022, 42 (4):  1631-1644.  DOI: 10.1007/s10473-022-0420-5
In this paper, we prove a Second Main Theorem for holomorphic mappings in a disk whose image intersects some families of nonlinear hypersurfaces (totally geodesic hypersurfaces with respect to a meromorphic connection) in the complex projective space $\mathbb{P}^k$. This is a generalization of Cartan's Second Main Theorem. As a consequence, we establish a uniqueness theorem for holomorphic mappings which intersect $O(k^3)$ many totally geodesic hypersurfaces.
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SOME FURTHER RESULTS FOR HOLOMORPHIC MAPS ON PARABOLIC RIEMANN SURFACES
Menglong HONG
Acta mathematica scientia,Series B. 2022, 42 (4):  1645-1665.  DOI: 10.1007/s10473-022-0421-4
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In [1], they generalized R. Nevanlinna's results to $Y$, where $Y$ is a parabolic Riemann Surface. In this paper, following their method, we develop some further results for holomorphic maps on $Y$, including the maps into ${\mathbb P}^n(\mathbb{C})$, the complex projective varieties, and Abelian varieties.
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THE GLOBAL COMBINED QUASI-NEUTRAL AND ZERO-ELECTRON-MASS LIMIT OF NON-ISENTROPIC EULER-POISSON SYSTEMS
Yongfu YANG, Qiangchang JU, Shuang ZHOU
Acta mathematica scientia,Series B. 2022, 42 (4):  1666-1680.  DOI: 10.1007/s10473-022-0422-3
We consider a non-isentropic Euler-Poisson system with two small parameters arising in the modeling of unmagnetized plasmas and semiconductors. On the basis of the energy estimates and the compactness theorem, the uniform global existence of the solutions and the combined quasi-neutral and zero-electron-mass limit of the system are proved when the initial data are close to the constant equilibrium state. In particular, the limit is rigorously justified as the two parameters tend to zero independently.
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THE METRIC GENERALIZED INVERSE AND ITS SINGLE-VALUE SELECTION IN THE PRICING OF CONTINGENT CLAIMS IN AN INCOMPLETE FINANCIAL MARKET
Zi WANG, Xiaoling WANG, Yuwen WANG
Acta mathematica scientia,Series B. 2022, 42 (4):  1681-1689.  DOI: 10.1007/s10473-022-0423-2
This article continues to study the research suggestions in depth made by M.Z. Nashed and G.F. Votruba in the journal "Bull. Amer. Math. Soc." in 1974. Concerned with the pricing of non-reachable "contingent claims" in an incomplete financial market, when constructing a specific bounded linear operator $A: l_1^n\rightarrow l_2$ from a non-reflexive Banach space $l_1^n$ to a Hilbert space $l_2$, the problem of non-reachable "contingent claims" pricing is reduced to researching the (single-valued) selection of the (set-valued) metric generalized inverse $A^\partial$ of the operator $A$. In this paper, by using the Banach space structure theory and the generalized inverse method of operators, we obtain a bounded linear single-valued selection $A^\sigma=A^+$ of $A^\partial$.
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TIME ANALYTICITY FOR THE HEAT EQUATION ON GRADIENT SHRINKING RICCI SOLITONS
Jiayong WU
Acta mathematica scientia,Series B. 2022, 42 (4):  1690-1700.  DOI: 10.1007/s10473-022-0424-1
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On a complete non-compact gradient shrinking Ricci soliton, we prove the analyticity in time for smooth solutions of the heat equation with quadratic exponential growth in the space variable. This growth condition is sharp. As an application, we give a necessary and sufficient condition on the solvability of the backward heat equation in a class of functions with quadratic exponential growth on shrinkers.
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