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1.
The article investigates the equation
$ {u_t}{\text{ = }}{\left( {u{u_x}} \right)_x}{\text{ + }}\left( {u - {u_0}} \right)\left( {u - {u_1}} \right){\text{,}}\quad \quad {u_1} > {u_0} > 0. $ {u_t}{\text{ = }}{\left( {u{u_x}} \right)_x}{\text{ + }}\left( {u - {u_0}} \right)\left( {u - {u_1}} \right){\text{,}}\quad \quad {u_1} > {u_0} > 0.  相似文献   

2.
In this work, we prove the Cauchy–Kowalewski theorem for the initial-value problem
$$\begin{aligned} \frac{\partial w}{\partial t}= & {} Lw \\ w(0,z)= & {} w_{0}(z) \end{aligned}$$
where
$$\begin{aligned} Lw:= & {} E_{0}(t,z)\frac{\partial }{\partial \overline{\phi }}\left( \frac{ d_{E}w}{dz}\right) +F_{0}(t,z)\overline{\left( \frac{\partial }{\partial \overline{\phi }}\left( \frac{d_{E}w}{dz}\right) \right) }+C_{0}(t,z)\frac{ d_{E}w}{dz} \\&+G_{0}(t,z)\overline{\left( \frac{d_{E}w}{dz}\right) } +A_{0}(t,z)w+B_{0}(t,z)\overline{w}+D_{0}(t,z) \end{aligned}$$
in the space \(P_{D}\left( E\right) \) of Pseudo Q-holomorphic functions.
  相似文献   

3.
We prove an estimate of Carleman type for the one dimensional heat equation $$ u_t - \left( {a\left( x \right)u_x } \right)_x + c\left( {t,x} \right)u = h\left( {t,x} \right),\quad \left( {t,x} \right) \in \left( {0,T} \right) \times \left( {0,1} \right), $$ where a(·) is degenerate at 0. Such an estimate is derived for a special pseudo-convex weight function related to the degeneracy rate of a(·). Then, we study the null controllability on [0, 1] of the semilinear degenerate parabolic equation $$ u_t - \left( {a\left( x \right)u_x } \right)_x + f\left( {t,x,u} \right) = h\left( {t,x} \right)\chi _\omega \left( x \right), $$ where (t, x) ∈(0, T) × (0, 1), ω=(α, β) ⊂⊂ [0, 1], and f is locally Lipschitz with respect to u. Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday  相似文献   

4.
On the Range of the Aluthge Transform   总被引:1,自引:0,他引:1  
Let be the algebra of all bounded linear operators on a complex separable Hilbert space For an operator let be the Aluthge transform of T and we define for all where T = U|T| is a polar decomposition of T. In this short note, we consider an elementary property of the range of Δ. We prove that R(Δ) is neither closed nor dense in However R(Δ) is strongly dense if is infinite dimensional. An erratum to this article is available at .  相似文献   

5.
Carleman estimates for one-dimensional degenerate heat equations   总被引:1,自引:0,他引:1  
In this paper, we are interested in controllability properties of parabolic equations degenerating at the boundary of the space domain. We derive new Carleman estimates for the degenerate parabolic equation $$ w_t + \left( {a\left( x \right)w_x } \right)_x = f,\quad \left( {t,x} \right) \in \left( {0,T} \right) \times \left( {0,1} \right), $$ where the function a mainly satisfies $$ a \in \mathcal{C}^0 \left( {\left[ {0,1} \right]} \right) \cap \mathcal{C}^1 \left( {\left( {0,1} \right)} \right),a \gt 0 \hbox{on }\left( {0,1} \right) \hbox{and }\frac{1} {{\sqrt a }} \in L^1 \left( {0,1} \right). $$ We are mainly interested in the situation of a degenerate equation at the boundary i.e. in the case where a(0)=0 and / or a(1)=0. A typical example is a(x)=xα (1 − x)β with α, β ∈ [0, 2). As a consequence, we deduce null controllability results for the degenerate one dimensional heat equation $$ u_t - (a(x)u_x )_x = h\chi _w ,\quad (t,x) \in (0,T) \times (0,1),\quad \omega \subset \subset (0,1). $$ The present paper completes and improves previous works [7, 8] where this problem was solved in the case a(x)=xα with α ∈[0, 2). Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday  相似文献   

6.
7.
Ukrainian Mathematical Journal - We prove the inequality$$ \overline{\omega}(t)\le \underset{s>0}{\operatorname{inf}}\left(\omega \left(\frac{s}{2}\right)+\frac{\omega (s)}{s}t\right), $$...  相似文献   

8.
We consider the Cauchy problem for the nonlinear dissipative evolution system with ellipticity on one dimensional space
with S. Q. Tang and H. Zhao [4] have considered the problem and obtained the optimal decay property for suitably small data. In this paper we derive the asymptotic profile using the Gauss kernel G(t, x), which shows the precise behavior of solution as time tends to infinity. In fact, we will show that the asymptotic formula
holds, where D0, β0 are determined by the data. It is the key point to reformulate the system to the nonlinear parabolic one by suitable changing variables. (Received: January 8, 2005)  相似文献   

9.
In this paper, the author proves the existence and uniqueness of nonnegative solution for the first boundary value problem of uniform degenerated parabolic equation $$\[\left\{ {\begin{array}{*{20}{c}} {\frac{{\partial u}}{{\partial t}} = \sum {\frac{\partial }{{\partial {x_i}}}\left( {v(u){A_{ij}}(x,t,u)\frac{{\partial u}}{{\partial {x_j}}}} \right) + \sum {{B_i}(x,t,u)} \frac{{\partial u}}{{\partial {x_i}}}} + C(x,t,u)u\begin{array}{*{20}{c}} {}&{(x,t) \in [0,T]} \end{array},}\{u{|_{t = 0}} = {u_0}(x),x \in \Omega ,}\{u{|_{x \in \partial \Omega }} = \psi (s,t),0 \le t \le T} \end{array}} \right.\]$$ $$\[\left( {\frac{1}{\Lambda }{{\left| \alpha \right|}^2} \le \sum {{A_{ij}}{\alpha _i}{\alpha _j}} \le \Lambda {{\left| \alpha \right|}^2},\forall a \in {R^n},0 < \Lambda < \infty ,v(u) > 0\begin{array}{*{20}{c}} {and}&{v(u) \to 0\begin{array}{*{20}{c}} {as}&{u \to 0} \end{array}} \end{array}} \right)\]$$ under some very weak restrictions, i.e. $\[{A_{ij}}(x,t,r),{B_i}(x,t,r),C(x,t,r),\sum {\frac{{\partial {A_{ij}}}}{{\partial {x_j}}}} ,\sum {\frac{{\partial {B_i}}}{{\partial {x_i}}} \in \overline \Omega } \times [0,T] \times R,\left| {{B_i}} \right| \le \Lambda ,\left| C \right| \le \Lambda ,\],\[\left| {\sum {\frac{{\partial {B_i}}}{{\partial {x_i}}}} } \right| \le \Lambda ,\partial \Omega \in {C^2},v(r) \in C[0,\infty ).v(0) = 0,1 \le \frac{{rv(r)}}{{\int_0^r {v(s)ds} }} \le m,{u_0}(x) \in {C^2}(\overline \Omega ),\psi (s,t) \in {C^\beta }(\partial \Omega \times [0,T]),0 < \beta < 1\],\[{u_0}(s) = \psi (s,0).\]$  相似文献   

10.
For the Neumann sinh-Gordon equation on the unit ball
we construct sequence of solutions which exhibit a multiple blow up at the origin, where λ ±  are positive parameters. It answers partially an open problem formulated in Jost et al. [Calc Var Partial Diff Equ 31(2):263–276]. The research of the first named author is supported by M. U. R. S. T., project “Variational methods and nonlinear differential equations”. The research of the second named author is supported by an Earmarked grant from RGC of Hong Kong.  相似文献   

11.
The purpose of this paper is to give characterizations for uniform exponential dichotomy of evolution families on the real line. We consider a general class of Banach function spaces denoted and we prove that if with and the pair is admissible for an evolution family then is uniformly exponentially dichotomic. By an example we show that the admissibility of the pair for an evolution family is not a sufficient condition for uniform exponential dichotomy. As applications, we deduce necessary and sufficient conditions for uniform exponential dichotomy of evolution families in terms of the admissibility of the pairs and with   相似文献   

12.
Lithuanian Mathematical Journal - We present upper bounds of the integral $$ {\int}_{-\infty}^{\infty }{\left|x\right|}^l\left|\mathrm{P}\left\{{Z}_N0\left({S}_N{X}_1+\dots +{X}_N\right) $$ of...  相似文献   

13.
In this paper we establish the existence of nontrivial solutions to
with V x superlinear in x.  相似文献   

14.
We consider the Cauchy problem for the nonlinear dissipative evolution system with ellipticity on one dimensional space
$ \left\{{{ll} {\psi_t=-\left({1-\alpha}\right)\psi-\theta_x+\alpha\psi_{xx},}&{\left( {t,x} \right) \in \left( {0,\infty } \right) \times {\bf R}}\\ {\theta _t = - \left( {1 - \alpha } \right)\theta + \nu ^2 \psi _x + \alpha \theta _{xx} + 2\psi \theta _x ,} } \right. $ \left\{{\begin{array}{ll} {\psi_t=-\left({1-\alpha}\right)\psi-\theta_x+\alpha\psi_{xx},}&{\left( {t,x} \right) \in \left( {0,\infty } \right) \times {\bf R}}\\ {\theta _t = - \left( {1 - \alpha } \right)\theta + \nu ^2 \psi _x + \alpha \theta _{xx} + 2\psi \theta _x ,} \end{array}} \right.  相似文献   

15.
Ru Ying  XUE 《数学学报(英文版)》2010,26(12):2421-2442
we study an initial-boundary-value problem for the "good" Boussinesq equation on the half line
{δt^2u-δx^2u+δx^4u+δx^2u^2=0,t〉0,x〉0.
u(0,t)=h1(t),δx^2u(0,t) =δth2(t),
u(x,0)=f(x),δtu(x,0)=δxh(x).
The existence and uniqueness of low reguality solution to the initial-boundary-value problem is proved when the initial-boundary data (f, h, h1, h2) belong to the product space
H^5(R^+)×H^s-1(R^+)×H^s/2+1/4(R^+)×H^s/2+1/4(R^+)
1 The analyticity of the solution mapping between the initial-boundary-data and the with 0 ≤ s 〈 1/2. solution space is also considered.  相似文献   

16.
Let{(t);t∈R_ ~N}be a d-dimensional N-parameter generalized Brownian sheet.Necessaryand sufficient conditions for a compact set E×F to be a polar set for(t,(t))are proved.It is also provedthat if 2N≤αd,then for any compact set ER_>~N,d-2/2 Dim E≤inf{dimF:F ∈ B(R~d),P{(E)∩F≠φ}>0}≤d-2/β DimE,and if 2N>αd,then for any compact set FR~d\{0},α/2(d-DimF)≤inf{dimE:E∈B(R_>~N),P{(E)∩F≠φ}>0}≤β/2(d-DimF),where B(R~d)and B(R_>~N)denote the Borel σ-algebra in R~d and in R_>~N respectively,dim and Dim are Hausdorffdimension and Packing dimension respectively.  相似文献   

17.
The initial boundary value problem
$ {*{20}{c}} {\rho {u_{tt}} - {{\left( {\Gamma {u_x}} \right)}_x} + A{u_x} + Bu = 0,} \hfill & {x > 0,\quad 0 < t < T,} \hfill \\ {u\left| {_{t = 0}} \right. = {u_t}\left| {_{t = 0}} \right. = 0,} \hfill & {x \geq 0,} \hfill \\ {u\left| {_{x = 0}} \right. = f,} \hfill & {0 \leq t \leq T,} \hfill \\ $ \begin{array}{*{20}{c}} {\rho {u_{tt}} - {{\left( {\Gamma {u_x}} \right)}_x} + A{u_x} + Bu = 0,} \hfill & {x > 0,\quad 0 < t < T,} \hfill \\ {u\left| {_{t = 0}} \right. = {u_t}\left| {_{t = 0}} \right. = 0,} \hfill & {x \geq 0,} \hfill \\ {u\left| {_{x = 0}} \right. = f,} \hfill & {0 \leq t \leq T,} \hfill \\ \end{array}  相似文献   

18.
There are lots of results on the solutions of the heat equation but much less on those of the Hermite heat equation due to that its coefficients are not constant and even not bounded. In this paper, we find an explicit relation between the solutions of these two equations, thus all known results on the heat equation can be transferred to results on the Hermite heat equation, which should be a completely new idea to study the Hermite equation. Some examples are given to show that known results on the Hermite equation are obtained easily by this method, even improved. There is also a new uniqueness theorem with a very general condition for the Hermite equation, which answers a question in a paper in Proc. Japan Acad. (2005). Supported partially by 973 project (2004CB318000)  相似文献   

19.
Summary LetX be a diffusion in natural scale on (0,1], with 1 reflecting, and letc(x)(H x ) andv(x)var (H x ), whereH x =inf{t: X t =x}. Let x =sup{t:X t =x}. The main results of this paper are firstly that (i)c is slowly varying; (ii) are all equivalent: and secondly that (v) are all equivalent, and are implied by the condition . Other partial results for more general limit theorems are proved, and new results on regular variation are established.  相似文献   

20.
We are concerned with existence, positivity property and long-time behavior of solutions to the following initial boundary value problem of a fourth order degenerate parabolic equation in higher space dimensions   相似文献   

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