|
$
u_{tt} = u_{xx} + i2u_{ttx} + u_{ttxx}
$
u_{tt} = u_{xx} + i2u_{ttx} + u_{ttxx}
相似文献
2.
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
3.
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
4.
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
5.
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
6.
We provide examples of solutions to parabolic problems with nontrivial blow-up sets of dimension strictly smaller than the space dimension. To this end we just consider different diffusion operators in different variables, for example, or . For both equations, we prove that there exists a solution that blows up in the segment . 相似文献
7.
Applying Nash-Moser's implicit function theorem, the author proves the existence of periodic solution to nonlinear wave equation u_{tt} - u_{xx} + εg(t, x, u, u_t, u_x, u_{tt}, u_{tx}, u_{xx}) = 0 with a dissipative boundary condition, provided ε is sufficiently small. 相似文献
8.
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
9.
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
10.
Based on the method of qualitative research in ordinary differential equations, lt is proved that, for any given positive β and ϒ, and for any given real a, b and c, the Burgers-KdV equation u_t + uu_x - ϒu_{xx} + βu_{xxx} = 0 has at least one, but at most finite Static solutions satisfying the same boundary conditions u(0, t) = a,u(1, t) = b \quad and u_x(1, t) = c on the interval [0, 1] of x. Some sufficient conditions on the global stability for certain statle solutions are given. 相似文献
11.
We discuss the existence of global classical solution for the uniformly parabolic equation
|
相似文献
12.
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
13.
Let u=u（x,t,uo）represent the global solution of the initial value problem for the one-dimensional fluid dynamics equation ut-εuxxt＋δux＋γHuxx＋βuxxx＋f（u）x=αuxx,u（x,0）=uo（x）, whereα〉0,β〉0,γ〉0,δ〉0 andε〉0 are constants.This equation may be viewed as a one-dimensional reduction of n-dimensional incompressible Navier-Stokes equations. The nonlinear function satisfies the conditions f（0）=0,｜f（u）｜→∞as ｜u｜→∞,and f∈C^1（R）,and there exist the following limits Lo=lim sup/u→o f（u）/u^3 and L∞=lim sup/u→∞ f（u）/u^5 Suppose that the initial function u0∈L^I（R）∩H^2（R）.By using energy estimates,Fourier transform,Plancherel＇s identity,upper limit estimate,lower limit estimate and the results of the linear problem vt-εv（xxt）＋δvx＋γHv（xx）＋βv（xxx）=αv（xx）,v（x,0）=vo（x）, the author justifies the following limits（with sharp rates of decay） lim t→∞[（1＋t）^（m＋1/2）∫｜uxm（x,t）｜^2dx]=1/2π（π/2α）^（1/2）m!!/（4α）^m[∫R uo（x）dx]^2, if∫R uo（x）dx≠0, where 0!!=1,1!!=1 and m!!=1·3…（2m-3）…（2m-1）.Moreover lim t→∞[（1＋t）^（m＋3/2）∫R｜uxm（x,t）｜^2dx]=1/2π（x/2α）^（1/2）（m＋1）!!/（4α）^（m＋1）[∫Rρo（x）dx]^2, if the initial function uo（x）=ρo′（x）,for some functionρo∈C^1（R）∩L^1（R）and∫Rρo（x）dx≠0. 相似文献
14.
§1.Ｉｎｔｒｏｄｕｃｔｉｏｎ ＳｉｎｃｅｔｈｅｄｉｓｃｏｖｅｒｙｂｙＫｏｒｔｅｗｅｇａｎｄｄｅＶｒｉｅｓ(1895)ａｎｄｔｈｅｗｏｒｋｏｆＺａｂｕｓｋｙａｎｄＫｒｕｓｔａｌ(1965),ｔｈｅｒｅｈａｖｅｂｅｅｎｎｕｍｅｒｏｕｓｓｉｇｎｉｆｉｃａｎｔｃｏｎｔｒｉｂｕｔｉｏｎｓｔｏｔｈｅＫｏｒｔｅｗｅｇｄｅＶｒｉｅｓ(ＫｄＶ)ｅｑｕａｔｉｏｎｓａｎｄｔｈｅｓｏｌｉｔｏｎｓｏｌｕｔｉｏｎｓ,ｅｓｐｅｃｉａｌｌｙｔｈｅｍｅｔｈｏｄｏｆＩｎｖｅｒｓｅＳｃａｔｔｅｒｉｎｇＴｒａｎｓｆｏｒｍａｎｄｔｈｅｍｅｔｈｏｄｏｆ… 相似文献
15.
The solvability of the nonlocal boundary value problem in a class of functions is investigated for a quasilinear parabolic equation. The solution uniqueness follows from the maximum principle. 相似文献
16.
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
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.
The following coupled Schrodinger system with a small perturbation is considered, where β and ε are small parameters. The whole system has a periodic solution with the aid of a Fourier series expansion technique, and its dominant system has a heteroclinic solution. Then adjusting some appropriate constants and applying the fixed point theorem and the perturbation method yield that this heteroclinic solution deforms to a heteroclinic solution exponentially approaching the obtained periodic solution （called the generalized heteroclinic solution thereafter）. 相似文献
19.
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
20.
ＦＩＮＩＴＥＤＩＦＦＥＲＥＮＣＥＳＣＨＥＭＥＳＯＦＴＨＥＮＯＮＬＩＮＥＡＲＰＳＥＵＤＯ－ＰＡＲＡＢＯＬＩＣＳＹＳＴＥＭＤＵＭＩＮＧＳＨＥＮＧ（杜明笙）（ＩｎｓｔｉｔｕｔｅｏｆＡｐｐｌｉｅｄＰｈｙｓｉｃｓａｎｄＣｏｍｐｕｔａｔｉｏｎａｌＭａｔｈｅｍａｔ... 相似文献
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