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By
Zhang, Yingshu; Li, Lang; Fang, Shaomei
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The Cauchy problem of the generalized KuramotoSivashinsky equation in multidimensions (n ≥ 3) is considered. Based on Green’s function method, some ingenious energy estimates are given. Then the global existence and pointwise convergence rates of the classical solutions are established. Furthermore, the L^{p} convergence rate of the solution is obtained.
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By
Hou, Xianming; Wu, Huoxiong
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In this paper, we establish the following limiting weaktype behaviors of LittlewoodPaley gfunction g': for nonnegative function f ∈ L^{1}(R^{n}),
$$\mathop {\lim }\limits_{\lambda \to {0_ + }} \lambda m\left( {\left\{ {x \in {\mathbb{R}^n}:g\varphi f\left( x \right) > \lambda } \right\}} \right) = m\left( {\left\{ {x \in {\mathbb{R}^n}:{{\left( {\int_0^\infty {{{\left {\varphi r\left( x \right)} \right}^2}\frac{{dr}}{r}} } \right)}^{1/2}} > 1} \right\}} \right){\left\ f \right\_1}$$
and
$$\mathop {\lim }\limits_{t \to {0_ + }} m\left( {\left\{ {x \in {\mathbb{R}^n}:g\varphi {f_t}\left( x \right)  {{\left( {\int_0^\infty {{{\left {\varphi r\left( x \right)} \right}^2}\frac{{dr}}{r}} } \right)}^{1/2}} > 1} \right\}} \right) = 0$$
, where f_{t}(x) = t^{−n}f(t^{−1}x) for t > 0. Meanwhile, the corresponding results for Marcinkiewicz integral and its fractional version with kernels satisfying L_{α}^{q} Dini condition are also given.
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By
Hou, Jinchuan; Zhao, Haili
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A property (C) for permutation pairs is introduced. It is shown that if a pair {π_{1}, π_{2}} of permutations of (1, 2, · · ·, n) has property (C), then the Dtype map
$${\Phi _{{\pi _{1,}}{\pi _2}}}$$
on n × n complex matrices constructed from {π_{1}, π_{2}} is positive. A necessary and sufficient condition is obtained for a pair {π_{1}, π_{2}} to have property (C), and an easily checked necessary and sufficient condition for the pairs of the form {π^{p}, π^{q}} to have property (C) is given, where π is the permutation defined by π(i) = i + 1 mod n and 1 ≤ p < q ≤ n.
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By
Xu, Lanxi; Li, Ziyi
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Nonlinear stability of the motionless doublediffusive solution of the problem of an infinite horizontal fluid layer saturated porous medium is studied. The layer is heated and salted from below. By introducing two balance fields and through defining new energy functionals it is proved that for CLe ≥ R, Le ≤ 1 the motionless doublediffusive solution is always stable and for CLe < R, Le < 1 the solution is globally exponentially and nonlinearly stable whenever R < 4π^{2}+LeC, where Le, C and R are the Lewis number, Rayleigh number for solute and heat, respectively. Moreover, the nonlinear stability proved here is global and exponential, and the stabilizing effect of the concentration is also proved.
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By
Wu, ShunTang
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In this paper, we consider the following viscoelastic wave equation with delay
$${\left {{u_t}} \right^\rho }{u_{tt}}  \Delta u  \Delta {u_{tt}} + \int_0^t {g\left( {t  s} \right)} \Delta u\left( s \right)ds + {\mu _1}{\mu _t}\left( {x,t} \right) + {\mu _2}{\mu _t}\left( {x,t  \tau } \right) = b{\left u \right^{p  2}}u$$
in a bounded domain. Under appropriate conditions on μ_{1}, μ_{2}, the kernel function g, the nonlinear source and the initial data, there are solutions that blow up in finite time.
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By
Chen, Zongxuan; Shon, Kwang Ho
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Consider the difference Riccati equation xxxx, where A, B, C, D are meromorphic functions, we give its solution family with oneparameter
$$H\left( {f\left( z \right)} \right) = \left\{ {{f_0}\left( z \right),f\left( z \right) = \frac{{\left( {{f_1}\left( z \right)  {f_0}\left( z \right)} \right)\left( {{f_2}\left( z \right)  {f_0}\left( z \right)} \right)}}{{Q\left( z \right)\left( {{f_2}\left( z \right)  {f_1}\left( z \right)} \right) + \left( {{f_2}\left( z \right)  {f_0}\left( z \right)} \right)}} + {f_0}\left( z \right)} \right\}$$
, where Q(z) is any constant in C or any periodic meromorphic function with period 1, and f_{0}(z), f_{1}(z), f_{2}(z) are its three distinct meromorphic solutions.
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By
Wang, Zejun; Zhang, Qi
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In this paper, we use LaxOleinik formula to study the asymptotic behavior for the initial problem of scalar conservation law u_{t} + F(u)x = 0. First, we prove a simple but useful property of LaxOleinik formula (Lemma 2.7). In fact, denote the Legendre transform of F(u) as L(σ), then we can prove that the quantity F(q)−qF′(q)+ L(F′(q)) is a constant independent of q. As a simple application, we first give the solution of Riemann problem without using of RankineHugoniot condition and entropy condition. Then we study the asymptotic behavior of the problem with some special initial data and prove that the solution contains only a single shock for t > T*. Meanwhile, we can give the equation of the shock and an explicit value of T*.
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By
Wang, Chaojun; Cui, Yanyan; Liu, Hao
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In this article, we mainly study the invariance of some biholomorphic mappings with special geometric characteristics under the extension operators. First we generalize the RoperSuffridge extension operators on BergmanHartogs domains. Then, by the geometric characteristics of subclasses of biholomorphic mappings, we conclude that the modified RoperSuffridge operators preserve the properties of S_{Ω}* (β,A,B), parabolic and spirallike mappings of type β and order ρ, strong and almost spirallike mappings of type β and order α as well as almost starlike mappings of complex order λ on
$$\Omega_{p_{1}}^{B^{n}},\ldots,_{p_{s},q}$$
under different conditions, respectively. The conclusions provide new approaches to construct these biholomorphic mappings in several complex variables.
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By
Li, Xiuwen; Liu, Zhenhai; Li, Jing; Tisdell, Chris
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In this paper, we are concerned with the existence of mild solution and controllability for a class of nonlinear fractional control systems with damping in Hilbert spaces. Our first step is to give the representation of mild solution for this control system by utilizing the general method of Laplace transform and the theory of (α, γ)regularized families of operators. Next, we study the solvability and controllability of nonlinear fractional control systems with damping under some suitable sufficient conditions. Finally, two examples are given to illustrate the theory.
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By
Geng, Shifeng; Tang, Yanjuan
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This article is involved with the asymptotic behavior of solutions for nonlinear hyperbolic system with external friction. The global existence of classical solutions is proven, and L^{p} convergence rates are obtained. Compared with the results obtained by Hsiao and Liu, better convergence rates are obtained in this article.
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