共查询到20条相似文献，搜索用时 15 毫秒
1.
In this paper, we consider the following nonlinear equation ut＋2kux-uxxt＋au^2ux=2uxuxx＋uuxxx,which is a modified form of the Camassa-Holm equation. We construct four new explicit periodic wave solutions by bifurcation method of dynamical systems. We also obtain two explicit solitary wave solutions via the limits of the explicit periodic wave solutions. One of the two solitary wave solutions is new. 相似文献
2.
We study the first boundary-value problem for the following mixed-type equation of the second kind:
|
$
u_{xx} + yu_{yy} + au_y - b^2 u = 0
$
u_{xx} + yu_{yy} + au_y - b^2 u = 0
相似文献
3.
The smoothness properties of weak solutions to the Dirichlet problem for m-Hessian equations are studied. Namely, fully nonlinear second-order equations of the form
|
$
tr_m u_{xx} = f^m
$
tr_m u_{xx} = f^m
相似文献
4.
Many problems in mathematical physics are reduced to one- or multidimensional initial and initial-boundary value problems
for, generally speaking, strongly nonlinear Sobolev-type equations. In this work, local and global classical solvability is
studied for the one-dimensional mixed problem with homogeneous Riquier-type boundary conditions for a class of semilinear
long-wave equations
|
$
U_{tt} \left( {t,x} \right) - U_{xx} \left( {t,x} \right) - \alpha U_{ttxx} \left( {t,x} \right) = F\left( {t,x,U\left( {t,x} \right),U_x \left( {t,x} \right),U_{xx} \left( {t,x} \right),U_t \left( {t,x} \right),U_{tx} \left( {t,x} \right),U_{txx} \left( {t,x} \right)} \right)
$
U_{tt} \left( {t,x} \right) - U_{xx} \left( {t,x} \right) - \alpha U_{ttxx} \left( {t,x} \right) = F\left( {t,x,U\left( {t,x} \right),U_x \left( {t,x} \right),U_{xx} \left( {t,x} \right),U_t \left( {t,x} \right),U_{tx} \left( {t,x} \right),U_{txx} \left( {t,x} \right)} \right)
相似文献
5.
We consider the problem on the stability of the Oskolkov equations
|
$
(\lambda - \lambda _0 )u_{jt} + u_{jtxx} = u_j u_{ix} - \nu u_{jxx} + \varepsilon _j
$
(\lambda - \lambda _0 )u_{jt} + u_{jtxx} = u_j u_{ix} - \nu u_{jxx} + \varepsilon _j
相似文献
6.
We shall consider the third order hyperbolic equation in [0, T] × Rx
where α ≥ 2, Β ≥ 1, η ≥ 0, λ ≥ 0, Σ ≥ 0, Μ ≥ 0, Ω ≥ 0 and θ ≥ 0 are integers. We prove that the Cauchy problem (1) is Gevrey well-posed. 相似文献
7.
A Gelerkin type method, based on trigonometric functions and Crank-Nicolson discretizations of the time variable, is applied to compute solutions of the initial boundary value problem associated with the equation $$\rho _0 u_{tt} = a_1 u_{xx} + a_2 u_x^2 u_{xx} + a_3 u_{xtx} + f,x \in [0,1],t > 0.$$ Error estimates are derived and a numerical example is presented. 相似文献
8.
The paper suggests some conditions on the lower order terms, which provide that the solution of the Dirichlet problem for
the general elliptic equation of the second order
|
$
\begin{gathered}
- \sum\limits_{i,j = 1}^n {\left( {a_{i j} \left( x \right)u_{x_i } } \right)_{x_j } + } \sum\limits_{i = 1}^n {b_i \left( x \right)u_{x_i } - } \sum\limits_{i = 1}^n {\left( {c_i \left( x \right)u} \right)_{x_i } + d\left( x \right)u = f\left( x \right) - divF\left( x \right), x \in Q,} \hfill \\
\left. u \right|_{\partial Q} = u_0 \in L_2 \left( {\partial Q} \right) \hfill \\
\end{gathered}
$
\begin{gathered}
- \sum\limits_{i,j = 1}^n {\left( {a_{i j} \left( x \right)u_{x_i } } \right)_{x_j } + } \sum\limits_{i = 1}^n {b_i \left( x \right)u_{x_i } - } \sum\limits_{i = 1}^n {\left( {c_i \left( x \right)u} \right)_{x_i } + d\left( x \right)u = f\left( x \right) - divF\left( x \right), x \in Q,} \hfill \\
\left. u \right|_{\partial Q} = u_0 \in L_2 \left( {\partial Q} \right) \hfill \\
\end{gathered}
相似文献
9.
This paper concerns the study of the numerical approximation for the following initialboundary value problem
|
$
\left\{ \begin{gathered}
u_t - u_{xx} = f\left( u \right), x \in \left( {0,1} \right), t \in \left( {0,T} \right), \hfill \\
u\left( {0,t} \right) = 0, u_x \left( {1,t} \right) = 0, t \in \left( {0,T} \right), \hfill \\
u\left( {x,0} \right) = u_0 \left( x \right), x \in \left[ {0,1} \right], \hfill \\
\end{gathered} \right.
$
\left\{ \begin{gathered}
u_t - u_{xx} = f\left( u \right), x \in \left( {0,1} \right), t \in \left( {0,T} \right), \hfill \\
u\left( {0,t} \right) = 0, u_x \left( {1,t} \right) = 0, t \in \left( {0,T} \right), \hfill \\
u\left( {x,0} \right) = u_0 \left( x \right), x \in \left[ {0,1} \right], \hfill \\
\end{gathered} \right.
相似文献
10.
We consider the three dimensional Cauchy problem for the Laplace equation uxx(x,y,z)+ uyy(x,y,z)+ uzz(x,y,z) = 0, x ∈ R,y ∈ R,0 z ≤ 1, u(x,y,0) = g(x,y), x ∈ R,y ∈ R, uz(x,y,0) = 0, x ∈ R,y ∈ R, where the data is given at z = 0 and a solution is sought in the region x,y ∈ R,0 z 1. The problem is ill-posed, the solution (if it exists) doesn't depend continuously on the initial data. Using Galerkin method and Meyer wavelets, we get the uniform stable wavelet approximate solution. Furthermore, we shall give a recipe for choosing the coarse level resolution. 相似文献
11.
In this paper, we consider the nonlinear Kirchhoff-type equation $
u_{tt} + M(\left\| {D^m u(t)} \right\|_2^2 )( - \Delta )^m u + \left| {u_t } \right|^{q - 2} u_t = \left| {u_t } \right|^{p - 2} u
$
u_{tt} + M(\left\| {D^m u(t)} \right\|_2^2 )( - \Delta )^m u + \left| {u_t } \right|^{q - 2} u_t = \left| {u_t } \right|^{p - 2} u
with initial conditions and homogeneous boundary conditions. Under suitable conditions on the initial datum, we prove that
the solution blows up in finite time. 相似文献
12.
We study large time asymptotic behavior of solutions to the periodic problem for the nonlinear damped wave equation
|
$ \left\{ {l} u_{tt}+2\alpha u_{t}-\beta u_{xx}=-\lambda \left| u\right| ^{\sigma}u,\text{ }x\in \Omega ,t >0 , \\ u(0,x)=\phi \left( x\right) ,\text{}u_{t}(0,x)=\psi \left( x\right) ,\text{ }x\in \Omega , \right. $ \left\{ \begin{array}{l} u_{tt}+2\alpha u_{t}-\beta u_{xx}=-\lambda \left| u\right| ^{\sigma}u,\text{ }x\in \Omega ,t >0 , \\ u(0,x)=\phi \left( x\right) ,\text{}u_{t}(0,x)=\psi \left( x\right) ,\text{ }x\in \Omega , \end{array} \right. 相似文献
13.
We consider the first-order Cauchy problem
|
$
\begin{gathered}
\partial _z u + a(z,x,D_x )u = 0,0 < z \leqslant Z, \hfill \\
u|_{z = 0} = u_0 , \hfill \\
\end{gathered}
$
\begin{gathered}
\partial _z u + a(z,x,D_x )u = 0,0 < z \leqslant Z, \hfill \\
u|_{z = 0} = u_0 , \hfill \\
\end{gathered}
相似文献
14.
In questo lavoro si considera il problema
|
相似文献
15.
The existence of global weak solutions to the periodic boundary problem or the initial value problem for the nonlinear Pseudo-hyperbolic equation u_(tt)-[a_1+a_2(u_x)~(2m)]u_(xx)-a_3u_(xxt)=f(x,t,u,u_x) is proved by the method of the vanishing of the additional diffusion terms, Leray-Schauder's fixedpoint argument and Sobolev's estimates,where m≥1 is a natural number and a_i>0(i=1,2,3)are constants. 相似文献
16.
We study large time asymptotics of solutions to the BBM–Burgers equation
. We are interested in the large time asymptotics for the case, when the initial data have an arbitrary size. Let the initial
data
, and
. Then we prove that there exists a unique solution
to the Cauchy problem for the BBM–Burgers equation. We also find the large time asymptotics for the solutions
To the memory of Professor Tsutomu Arai
Submitted: February 5, 2006. Accepted: June 17, 2006. 相似文献
17.
On the basis of properties of the Vejvoda-Shtedry operator, we obtain solvability conditions for the 2π-periodic problem .
Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 6, pp. 818–821, June, 1998. 相似文献
18.
The third-order nonlinear dispersion PDE, as the key model,
is studied. Two Riemann’s problems for (0.1) with the initial data S
∓( x) = ∓ sgn. x create shock ( u( x, t) ≡ S
−( x)) and smooth rarefaction (for the data S
+) waves (see [16]). The concept of “δ-entropy” solutions and others are developed for establishing the existence and uniqueness
for (0.1) by using stable smooth δ-deformations of shock-type solutions. These are analogous to entropy theory for scalar
conservation laws such as u
t
+ uu
x
= 0, which were developed by Oleinik and Kruzhkov (in x ∊ ℝ
N
) in the 1950s–1960s. The Rosenau-Hyman K(2, 2) (compacton) equation which has a special importance for applications, is studied. Compactons as compactly supported travelling wave solutions
are shown to be δ-entropy. Shock and rarefaction waves are discussed for other NDEs such as .
This article was submitted by the author in English.
Dedicated to the memory of Professors O.A. Oleinik and S.N. Kruzhkov 相似文献
19.
Stochastic homogenization (with multiple fine scales) is studied for a class of nonlinear monotone eigenvalue problems. More
specifically, we are interested in the asymptotic behaviour of a sequence of realizations of the form
|
$
- div\left( {a\left( {T_1 \left( {\frac{x}
{{\varepsilon _1 }}} \right)\omega _1 ,T_2 \left( {\frac{x}
{{\varepsilon _2 }}} \right)\omega _2 ,\nabla u_\varepsilon ^\omega } \right)} \right) = \lambda _\varepsilon ^\omega \mathcal{C}\left( {u_\varepsilon ^\omega } \right)
$
- div\left( {a\left( {T_1 \left( {\frac{x}
{{\varepsilon _1 }}} \right)\omega _1 ,T_2 \left( {\frac{x}
{{\varepsilon _2 }}} \right)\omega _2 ,\nabla u_\varepsilon ^\omega } \right)} \right) = \lambda _\varepsilon ^\omega \mathcal{C}\left( {u_\varepsilon ^\omega } \right)
相似文献
20.
We study the behavior of positive solutions of the following Dirichlet problem
|
$
\left \{ {ll} -\Delta_{p}u=\lambda u^{s-1}+u^{q-1} &\quad {\rm in}
\enspace \Omega \\ u_{\mid\partial \Omega}=0
\right. $
\left \{ \begin{array}{ll} -\Delta_{p}u=\lambda u^{s-1}+u^{q-1} &\quad {\rm in}
\enspace \Omega \\ u_{\mid\partial \Omega}=0 \end{array}
\right. 相似文献
|
|
|
| | | | | | | | |
