共查询到20条相似文献,搜索用时 562 毫秒
1.
I. M. Nikol’skii 《Computational Mathematics and Modeling》2009,20(4):339-347
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 相似文献
$$\begin{aligned} \frac{\partial w}{\partial t}= & {} Lw \\ w(0,z)= & {} w_{0}(z) \end{aligned}$$ $$\begin{aligned} Lw:= & {} E_{0}(t,z)\frac{\partial }{\partial \overline{\phi }}\left( \frac{ d_{E}w}{dz}\right) +F_{0}(t,z)\overline{\left( \frac{\partial }{\partial \overline{\phi }}\left( \frac{d_{E}w}{dz}\right) \right) }+C_{0}(t,z)\frac{ d_{E}w}{dz} \\&+G_{0}(t,z)\overline{\left( \frac{d_{E}w}{dz}\right) } +A_{0}(t,z)w+B_{0}(t,z)\overline{w}+D_{0}(t,z) \end{aligned}$$ 3.
Carleman estimates for degenerate parabolic operators with applications to null controllability 总被引:1,自引:0,他引:1
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.
Kenji Nishihara 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2006,57(4):604-614
We consider the Cauchy problem for the nonlinear dissipative evolution system with ellipticity on one dimensional space
9.
Dong Guangchang 《数学年刊B辑(英文版)》1986,7(3):277-302
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.
Pierpaolo Esposito Juncheng Wei 《Calculus of Variations and Partial Differential Equations》2009,34(3):341-375
For the Neumann sinh-Gordon equation on the unit ball
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
14.
Kenji Nishihara 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2006,41(6):604-614
We consider the Cauchy problem for the nonlinear dissipative evolution system with ellipticity on one dimensional space
|