共查询到20条相似文献,搜索用时 812 毫秒
1.
The quasilinear parabolic equation $$\frac{{\partial u}}{{\partial t}} = \frac{\partial }{{\partial x}}\left( {\left| {\frac{{\partial u^k }}{{\partial x}}} \right|^{n - 1} \frac{{\partial u^k }}{{\partial x}}} \right) - \gamma u^m , k,\gamma > 0, m \geqslant 0, kn > 1$$ can be regarded as the generalized form of many well-known transport equations with transport coefficients that depend on the transported quantity. For example, the case n = 1 corresponds to heat transport in a medium with thermal conductivity and sinks which depend on the temperature in accordance with a power law [1, 2]; the case k = m = 1 describes the flow of a conducting non-Newtonian fluid in a transverse magnetic field [3]; the case k = 2, m = 0 corresponds to the magnetohydrodynamic flow of the same fluid in a transverse magnetic field in a laminar boundary layer [4]. In the general case, Eq. (0.1) describes turbulent flow in a porous medium with nonlinear sinks [5]. A characteristic feature of the processes described by Eq. (0.1) is the possibility of existence of a front x = xf(t) which strictly separates regions with u(x, t) = 0 from regions with u(x, t) > 0 in which the perturbations of the transported quantity are localized [6]. In the present paper, the change in the region of localization of the perturbations of the transported quantity is investigated in the Cauchy problems for Eq. (0.1). 相似文献
2.
Begoña Barrios Ireneo Peral Fernando Soria Enrico Valdinoci 《Archive for Rational Mechanics and Analysis》2014,213(2):629-650
The main goal of this work is to prove that every non-negative strong solution u(x, t) to the problem $$u_t + (-\Delta)^{\alpha/2}{u} = 0 \,\, {\rm for} (x, t) \in {\mathbb{R}^n} \times (0, T ), \, 0 < \alpha < 2,$$ can be written as $$u(x, t) = \int_{\mathbb{R}^n} P_t (x - y)u(y, 0) dy,$$ where $$P_t (x) = \frac{1}{t^{n/ \alpha}}P \left(\frac{x}{t^{1/ \alpha}}\right),$$ and $$P(x) := \int_{\mathbb{R}^n} e^{i x\cdot\xi-|\xi |^\alpha} d\xi.$$ This result shows uniqueness in the setting of non-negative solutions and extends some classical results for the heat equation by Widder in [15] to the nonlocal diffusion framework. 相似文献
3.
We prove the existence of solutions of various boundary-value problems for nonautonomous Hamiltonian systems with forcing terms $$\begin{gathered} \dot x(t) = H'_p (t, x(t), p(t)) + g(t), \hfill \\ \dot p(t) = - H'_x (t, x(t), p(t)) - f(t). \hfill \\ \end{gathered} $$ Among these problems is the existence of T-periodic solutions, namely those satisfying x(t+T)=x(t) and p(t+T)+p(t). A special study is made of the classical case, where H(x, p)=1/2 |p|2+V(x). In the case of parametric oscillations, where (f, g)=(0, 0) and t ? H(t, x, p) is T-periodic, we give a lower bound for the true (minimal) period of the T-periodic solution (x, p) produced by our method, and we prove the existence of an infinite number of subharmonics. 相似文献
4.
Summary Sufficient conditions are given for the stability and instability of the equilibrium position x=y=z=0 in the mechanical system
consisting of a material point constrained to move on the moving surface z=−λ(t)(x2+y2) (λ(t)>0) in a constant field of gravity (the axis 0z is directed vertically upward) under the action of viscous friction
of total dissipation.
Sommario Si danno condizioni sufficienti per la stabilità e la instabilità della posizione di equilibrio x=y=z=0 nel sistema meccanico che consiste di un punto materiale vincolato a muoversi sulla superficie mobile z=−λ(t)(x2+y2) (λ(t)>0) in un campo di gravità costante (l'asse 0z è diretto verticalmente e orientato verso l'alto) sotto l'azione di attriti viscosi con dissipazione completa.相似文献
5.
The differential equation considered is \(y'' - xy = y|y|^\alpha \) . For general positive α this equation arises in plasma physics, in work of De Boer & Ludford. For α=2, it yields similarity solutions to the well-known Korteweg-de Vries equation. Solutions are sought which satisfy the boundary conditions (1) y(∞)=0 (2) (1) $$y{\text{(}}\infty {\text{)}} = {\text{0}}$$ (2) $$y{\text{(}}x{\text{) \~( - }}\tfrac{{\text{1}}}{{\text{2}}}x{\text{)}}^{{{\text{1}} \mathord{\left/ {\vphantom {{\text{1}} \alpha }} \right. \kern-\nulldelimiterspace} \alpha }} {\text{ as }}x \to - \infty $$ It is shown that there is a unique such solution, and that it is, in a certain sense, the boundary between solutions which exist on the whole real line and solutions which, while tending to zero at plus infinity, blow up at a finite x. More precisely, any solution satisfying (1) is asymptotic at plus infinity to some multiple kA i(x) of Airy's function. We show that there is a unique k*(α) such that when k=k*(α) the condition (2) is also satisfied. If 0 *, the solution exists for all x and tends to zero as x→-∞, while if k>k * then the solution blows up at a finite x. For the special case α=2 the differential equation is classical, having been studied by Painlevé around the turn of the century. In this case, using an integral equation derived by inverse scattering techniques by Ablowitz & Segur, we are able to show that k*=1, confirming previous numerical estimates. 相似文献
6.
Consider the perturbed nonautonomous linear delay differential equation x(t) = - a(t)x(t-τ) + F(t, x1, t ⩾ 0 where x1(s)=x(t+s) for −δ≤s≤0. Suppose that a(t) ∈ C([0,∞), (0,∞)), τ≥0,F:[0, ∞) x C[−δ,0] → R is a continuous functions and F(t,0)
≡ 0. Here C[−δ,0] is the space of continuous functions Φ: [−δ,0] → R with ∥Φ∥<H for the norm | Φ |, where |·| is any norm
in R and 0<H≤+∞.
Most of the known papers [1–5,7] have been concerned with the local or global asymptotic behavior of the zero solution of
Eq. (*) when a(t) is independent of t i. e., a(t) is autonomous. The aim in this paper is to derive the sufficient conditions
for the global attractivity of the zero solution of of Eq. (*) When a(t) is nonautomous. Our results, which extend and improve
the known results, are even “sharp”. At the same time, the method used in this paper can be applicable to the perturbed nonlinear
equation.
Project supported by the Natural Science Foundation of Hunan 相似文献
7.
W. Czernous 《Nonlinear Oscillations》2011,13(4):595-612
We consider the initial boundary-value problem for a system of quasilinear partial functional differential equations of the
first order
$ {*{20}{c}} {{\partial_t}{z_i}\left( {t,x} \right) + \sum\limits_{j = 1}^n {{\rho_{ij}}\left( {t,x,V\left( {z;t,x} \right)} \right){\partial_{{x_j}}}{z_i}\left( {t,x} \right) = {G_i}\left( {t,x,V\left( {z;t,x} \right)} \right),} } \hfill & {1 \leq i \leq m,} \hfill \\ $ \begin{array}{*{20}{c}} {{\partial_t}{z_i}\left( {t,x} \right) + \sum\limits_{j = 1}^n {{\rho_{ij}}\left( {t,x,V\left( {z;t,x} \right)} \right){\partial_{{x_j}}}{z_i}\left( {t,x} \right) = {G_i}\left( {t,x,V\left( {z;t,x} \right)} \right),} } \hfill & {1 \leq i \leq m,} \hfill \\ \end{array} 相似文献
8.
We study questions of existence, uniqueness and asymptotic behaviour for the solutions of u(x, t) of the problem $$\begin{gathered} {\text{ }}u_t - \Delta u = \lambda e^u ,{\text{ }}\lambda {\text{ > 0, }}t > 0,{\text{ }}x{\text{ }}\varepsilon B, \hfill \\ (P){\text{ }}u(x,0) = u_0 (x),{\text{ }}x{\text{ }}\varepsilon B, \hfill \\ {\text{ }}u(x,t) = 0{\text{ }}on{\text{ }}\partial B \times (0,\infty ), \hfill \\ \end{gathered} $$ where B is the unit ball $\{ x\varepsilon R^N :|x|{\text{ }} \leqq {\text{ }}1\} {\text{ and }}N \geqq 3$ . Our interest is focused on the parameter λ 0=2(N?2) for which (P) admits a singular stationary solution of the form $$S(x) = - 2log|x|$$ . We study the dynamical stability or instability of S, which depends on the dimension. In particular, there exists a minimal bounded stationary solution u which is stable if $3 \leqq N \leqq 9$ , while S is unstable. For $N \geqq 10$ there is no bounded minimal solution and S is an attractor from below but not from above. In fact, solutions larger than S cannot exist in any time interval (there is instantaneous blow-up), and this happens for all dimensions. 相似文献
9.
杨作东 《应用数学和力学(英文版)》1996,17(5):465-476
EXISTENCEOFPOSITIVESOLUTIONSFORACLASSOFSINGULARTWOPOINTBOUNDARYVALUEPROBLEMSOFSECONDORDERNONLINEAREQUATION(杨作东)EXISTENCEOFPOS... 相似文献
10.
Enzo Vitillaro 《Archive for Rational Mechanics and Analysis》1999,149(2):155-182
We study abstract evolution equations with nonlinear damping terms and source terms, including as a particular case a nonlinear wave equation of the type $ \ba{cl} u_{tt}-\Delta u+ b|u_t|^{m-2}u_t=c|u|^{p-2}u, &;(t,x)\in [0,T)\times\Omega,\\[6pt] u(t,x)=0, &;(t,x)\in [0,T)\times\partial \Omega,\\[6pt] u(0,\cdot)=u_0\in H_0^1(\Omega), \quad u_t(0,\cdot)=v_0\in L^2(\Omega),\es&; \ea $ \ba{cl} u_{tt}-\Delta u+ b|u_t|^{m-2}u_t=c|u|^{p-2}u, &;(t,x)\in [0,T)\times\Omega,\\[6pt] u(t,x)=0, &;(t,x)\in [0,T)\times\partial \Omega,\\[6pt] u(0,\cdot)=u_0\in H_0^1(\Omega), \quad u_t(0,\cdot)=v_0\in L^2(\Omega),\es&; \ea where 0 < T £ ¥0
11.
12.
E. M. Prokhorov 《Fluid Dynamics》1987,22(4):652-656
The case of an infinitely slender wing that slightly disturbs a supersonic ideal gas flow is considered. The plan form and the free-stream Mach number M are given. The optimum surface of the wing y=g(x, z) is determined as a result of finding a bounded function of the local angles of attack M=g(x, z)/x that minimizes the drag coefficient cx for given values of the lift coefficient cy and the pitching moment coefficient mz. The problem is solved in the class of piecewise-constant functions for wings of complex geometry [1].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 185–189, July–August, 1987. 相似文献
13.
Zhi-Cheng Wang Wan-Tong Li Shigui Ruan 《Journal of Dynamics and Differential Equations》2008,20(3):573-607
In this paper, we study the existence, uniqueness and stability of traveling wave fronts in the following nonlocal reaction–diffusion
equation with delay
14.
We consider the second Painlevé transcendent $$\frac{{d^2 y}}{{dx^2 }} = xy + 2y^3 .$$ It is known that if y(x) ~ k Ai (x) as x → + ∞, where ?1<k<1 and Ai (x) denotes Airy's function, then $$y(x) \sim d|x|^{ - \tfrac{1}{4}} sin\{ \tfrac{2}{3}|x|^{\tfrac{3}{2}} - \tfrac{3}{4}d^2 1n|x| - c\} ,$$ where the constants d, c depend on k. This paper shows that $$d^2 = \pi ^{ - 1} 1n(1 - k^2 )$$ , which confirms a conjecture by Ablowitz & Segur. 相似文献
15.
We investigate the asymptotic behavior of solutions of the damped nonlinear oscillator equation
16.
R. D. Gregory 《Journal of Elasticity》1980,10(1):57-80
For the problem of bending of a semi-infinite strip x0, –1y1, with the sides y=±1 clamped, we give a proof that the end-data% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcea% qabeaarmWu51MyVXgaiuGacqWFgpGzdaWgaaWcbaGaaeiEaiaabIha% aeqaaGqbaOGae4hiaaIaaiikaiaaicdacaGGSaGae4hiaaIaamyEai% aacMcacqGFGaaicqGH9aqpcqGFGaaicaWGMbGaaiikaiaadMhacaGG% PaGaaiilaaqaaiab-z8aMnaaBaaaleaacaqG5bGaaeyEaaqabaGccq% GFGaaicaGGOaGaaGimaiaacYcacqGFGaaicaWG5bGaaiykaiab+bca% Giabg2da9iab+bcaGiaadAgacaGGOaGaamyEaiaacMcacaGGSaaaaa% a!5D6D!\[\begin{array}{l} \phi _{{\rm{xx}}} (0, y) = f(y), \\ \phi _{{\rm{yy}}} (0, y) = f(y), \\ \end{array}\] where f(y), g(y) are arbitrary independent functions prescribed on (–1,1), may be expanded as a series of the bi-orthogonal Papkovich-Fadle eigenfunctions for the strip. This represents an advance on the standard work of R. T. C. Smith [6], who proved such an expansion, but under conditions which are often not satisfied in practice. In particular we are able to solve this bi-harmonic boundary value problem when f, g do not satisfy the side conditions% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcea% qabeaacaWGMbGaaiikaiabgglaXkaaigdacaGGPaqedmvETj2BSbac% faGae8hiaaIaeyypa0Jae8hiaaIaamOzamaaCaaaleqabaGaai4jaa% aakiab-bcaGiaacIcacqGHXcqScaaIXaGaaiykaiab-bcaGiabg2da% 9iab-bcaGiaaicdacaGGSaaabaGaam4zaiaacIcacqGHXcqScaaIXa% Gaaiykaiab-bcaGiabg2da9iab-bcaGiaadEgadaahaaWcbeqaaiaa% cEcaaaGccqWFGaaicaGGOaGaeyySaeRaaGymaiaacMcacqWFGaaicq% GH9aqpcqWFGaaicaaIWaGaaiilaaaaaa!6222!\[\begin{array}{l} f( \pm 1) = f^' ( \pm 1) = 0, \\ g( \pm 1) = g^' ( \pm 1) = 0, \\ \end{array}\]and when the conditions of consistency% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% Waa8qmaeaacaWGNbGaaiikaiaadMhacaGGPaqedmvETj2BSbacfaGa% e8hiaaIaamizaiaadMhacqWFGaaicqWF9aqpcqWFGaaidaWdXaqaai% aadMhacaWGNbGaaiikaiaadMhacaGGPaGae8hiaaIaamizaiaadMha% cqWFGaaicqGH9aqpcqWFGaaicaaIWaaaleaacqWFsislcqWFXaqmae% aacqWFXaqma0Gaey4kIipaaSqaaiabgkHiTiaaigdaaeaacaaIXaaa% niabgUIiYdaaaa!5A1B!\[\int_{ - 1}^1 {g(y) dy = \int_{ - 1}^1 {yg(y) dy = 0} } \]are not satisfied.The present completeness proof thus answers questions raised recently (in the mathematically equivalent context of Stokes flow) by Joseph [3], and Joseph and Sturges [5], who showed that if the side conditions (A), (B) are relaxed then the corresponding eigenfunction series may still converge; but they left open the more difficult question of whether these series still converge to the data.The method of proof used here also succeeds in proving a corresponding completeness theorem for the Williams eigenfunctions for the wedge with the data.% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcea% qabeaadaabciqaamaalaaabaGaeyOaIylabaGaeyOaIyRaamOCaaaa% daqadiqaamaalaaabaGaaGymaaqaaiaadkhaaaqedmvETj2BSbacfi% Gae8NXdygacaGLOaGaayzkaaaacaGLiWoadaWgaaWcbaGaamOCaiab% g2da9iaaigdaaeqaaGqbaOGae4hiaaIaeyypa0Jae4hiaaIaamOzai% aacIcacqaH4oqCcaGGPaGaaiilaaqaamaaeiGabaWaaSaaaeaacqGH% ciITdaahaaWcbeqaaiaaikdaaaGccqaHgpGzaeaacqGHciITcqaH4o% qCdaahaaWcbeqaaiaaikdaaaaaaOWaaeWaceaadaWcaaqaaiaaigda% aeaacaWGYbaaaiab-z8aMbGaayjkaiaawMcaaaGaayjcSdWaaSbaaS% qaaiaadkhacqGH9aqpcaaIXaaabeaakiab+bcaGiabg2da9iab+bca% GiaadEgacaGGOaGaeqiUdeNaaiykaiaacYcaaaaa!6B9C!\[\begin{array}{l} \left. {\frac{\partial }{{\partial r}}\left( {\frac{1}{r}\phi } \right)} \right|_{r = 1} = f(\theta ), \\ \left. {\frac{{\partial ^2 \phi }}{{\partial \theta ^2 }}\left( {\frac{1}{r}\phi } \right)} \right|_{r = 1} = g(\theta ), \\ \end{array}\]prescribed on –<<, (where 2 is the wedge angle).Department of Mathematics, University of ManchesterOn leave of absence at the University of British Columbia, Vancouver, B.C. Canada, during 1977–79. This work was supported in part by N.R.C. grants Nos. A 9259 and A9117. 相似文献
17.
Xinyu He 《Journal of Mathematical Fluid Mechanics》2004,6(4):389-404
A self-similar solution of the three-dimensional (3d) incompressible Euler equations is defined byu(x,t) =U(y)/t*-t)
α, y = x/(t* ~ t)β,α,β> 0, whereU(y) satisfiesζU + βy. ΔU + U. VU + VP = 0,divU = 0. For α = β = 1/2, which is the limiting case of Leray’s self-similar Navier—Stokes equations, we prove the existence of(U,P) ε H3(Ω,R3 X R) in a smooth bounded domain Ω, with the inflow boundary data of non-zero vorticity. This implies the possibility that
solutions of the Euler equations blow up at a timet = t*, t* < +∞. 相似文献
18.
Ref. [1] discussed the existence of positive solutions of quasilinear two-point boundary problems: but it restricts O相似文献
19.
In this paper, we study real solutions of the nonlinear Helmholtz equation $$- \Delta u - k^2 u = f(x,u),\quad x\in \mathbb{R}^N$$ satisfying the asymptotic conditions $$u(x)=O\left(|x|^{\frac{1-N}{2}}\right) \quad {\rm and} \quad \frac{\partial^2 u}{\partial r^2}(x)+k^2u(x)=o\left(|x|^{\frac{1-N}{2}}\right) \quad {\rm as}\, r=|x| \to\infty.$$ We develop the variational framework to prove the existence of nontrivial solutions for compactly supported nonlinearities without any symmetry assumptions. In addition, we consider the radial case in which, for a larger class of nonlinearities, infinitely many solutions are shown to exist. Our results give rise to the existence of standing wave solutions of corresponding nonlinear Klein–Gordon equations with arbitrarily large frequency. 相似文献
20.
In this work, positive solutions to a doubly nonlinear parabolic equation with a nonlinear boundary condition are considered.
We study the problem where 0 < m, r, α < ∞ are parameters. It is known that for some values of the parameters there are solutions that blow up in finite time.
We determine in terms of α ,m, r the blow-up sets for these solutions. We prove that single point blow-up occurs if
max{m, r} < α,
global blow-up appears for the range of parameters
0 < m < α < r
and regional blow-up takes place if
0 < m < α = r and .
In this case the blow-up set consists of the interval . 相似文献
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