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1.
The aim of this paper is to discuss the Cauchy problem for degenerate quasilinear hyperbolic equations of the form \frac{∂u}{∂t} + \frac{∂u^m}{∂x} = -u^p, m > 1, p > 0 with measures as initial conditions. The existence and uniqueness of solutions are obtained. In particular, we prove the following results: (1) 0 < p < 1 is a necessary and sufficient condition for the above equations to have extinction property; (2) 0 < p < m is a necessary and sufficient condition for the above equations to have localization property of the propagation of perturbations.  相似文献   

2.
This paper deals with the following semilinear parabolic equations with nonlinear boundary conditions u_t - Δu = f(u) - λu,x ∈ Ω, t > 0 \frac{∂u}{∂n} = g(u), \qquad x ∈ ∂Ω, t > 0 It is proved that every positive equilibrium solution is a threshold.  相似文献   

3.
In this paper we consider the two-dimensional Muskat free boundary problem: Δu_i(x,t) = 0 in space-time domain Q_i (i = 1,2), here tis a parameter. The unknown surface Γ_pT (free boundary) is tltc common part of the boundaries of Q_1 and Q_2. The free boundary conditions are u_1(x,t) = u_2(x,t) and -k_1\frac{∂u_1}{∂n} = -k_2\frac{∂u_2}{∂n} = V_n. If the initial normal velocity of the free boundary is positive, we shall prove the existence of classical solution locally in time and uniqueness by making use of Newton's iteration method.  相似文献   

4.
Sufficient conditions for existence and nonuniqueness of radially symmetric solutions to the Robin boundary problem of the form Δu + a(||x||)|u|^{-p} = 0 \qquad in B ⊂ R^N \frac{∂u}{∂n} + λu = -α \qquad on ∂B are obtained.  相似文献   

5.
This paper deals with the global existence and blow-up of positive solutions to the systems: u_t = ∇(u^∇u) + u¹ + v^a v_t = ∇(v^n∇v) + u^b + v^k in B_R × (0, T) \frac{∂u}{∂η} = u^αv^p, \frac{∂v}{∂η} = u^qv^β on S_R × (0, T) u(x, 0) = u_0(x), v(x, 0} = v_0(x) in B_R We prove that there exists a global classical positive solution if and only if l ≤ l, k ≤ 1, m + α ≤ 1, n + β ≤ 1, pq ≤ (1 - m - α)(1 - n - β),ab ≤ 1, qa ≤ (1 - n - β) and pb ≤ (1 - m - α).  相似文献   

6.
In this paper, we study the general difference schemes with nonuniform meshes for the following problem: u_t = A(x,t,u,u_x)u_{xx}, + f(x,t,u,u_x), 0 < x < l, 0 < t ≤ T \qquad (1) u(0,t) = u(l ,t) = 0, 0 < t ≤ T \qquad\qquad (2) u(x,0) = φ(x), 0 ≤ x ≤ l \qquad\qquad (3) where u, φ, and f are m-dimensional vector valued functions, u_t = \frac{∂u}{∂t}, u_x = \frac{∂u}{∂x}, u_{xx} = \frac{∂²u}{∂_x²}. In the practical computation, we usually use the method of iteration to calculate the approximate solutions for the nonlinear difference schemes. Here the estimates of the iterative sequence constructed from the iterative difference schemes for the problem (1)-(3) is proved. Moreover, when the coefficient matrix A = A(x, t, u) is independent of u_x, t he convergence of the approximate difference solution for the iterative difference schemes to the unique solution of the problem (1)-(3) is proved without imposing the assumption of heuristic character concerning the existence of the unique smooth solution for the original problem (1)-(3).  相似文献   

7.
In this paper, the existence of travelling wave solution for nonlinear equation wiili non local advection ρ\frac{∂}{∂t}(\frac{u^m}{m}) = \frac{∂²u}{∂x²}-\frac{∂}{∂x}[φ(k∗u)u]+u^nf(u) is studied in the case of m ≥ 1, n ≥ 1. When ε,φ, f, m and n satisfy some determinate conditions, there exists the travelling wave solution.  相似文献   

8.
In this paper, the authors consider the positive solutions of the system of the evolution $p$-Laplacian equations $$\begin{cases} u_t ={\rmdiv}(| ∇u |^{p−2} ∇u) + f(u, v), & (x, t) ∈ Ω × (0, T ), & \\ v_t = {\rmdiv}(| ∇v |^{p−2} ∇v) + g(u, v), &(x, t) ∈ Ω × (0, T) \end{cases}$$with nonlinear boundary conditions $$\frac{∂u}{∂η}= h(u, v), \frac{∂v}{∂η} = s(u, v),$$and the initial data $(u_0, v_0)$, where $Ω$ is a bounded domain in$\boldsymbol{R}^n$with smooth boundary $∂Ω, p > 2$, $h(· , ·)$ and $s(· , ·)$ are positive $C^1$ functions, nondecreasing in each variable. The authors find conditions on the functions $f, g, h, s$ that prove the global existence or finite time blow-up of positive solutions for every $(u_0, v_0)$.  相似文献   

9.
Let $P(∆)$ be a polynomial of the Laplace operator $$∆ = \sum\limits^n_{j=1}\frac{∂^2}{∂x^2_j} \ \ on \ \ \mathbb{R}^n.$$ We prove the existence of a bounded right inverse of the differential operator $P(∆)$ in the weighted Hilbert space with the Gaussian measure, i.e., $L^2(\mathbb{R}^n ,e^{−|x|^2}).$  相似文献   

10.
In this paper, we consider the Cauchy problem \frac{∂u}{∂t} = Δφ(u) in R^N × (0, T] u(x,0} = u_0(x) in R^N where φ ∈ C[0,∞) ∩ C¹(0,∞), φ(0 ) = 0 and (1 - \frac{2}{N})^+ < a ≤ \frac{φ'(s)s}{φ(s)} ≤ m for some a ∈ ((1 - \frac{2}{n})^+, 1), s > 0. The initial value u_0 (z) satisfies u_0(x) ≥ 0 and u_0(x) ∈ L¹_{loc}(R^N). We prove that, under some further conditions, there exists a weak solution u for the above problem, and moreover u ∈ C^{α, \frac{α}{2}}_{x,t_{loc}} (R^N × (0, T]) for some α > 0.  相似文献   

11.
In this paper we study fully nonlinear elliptic equations F(D²u, x) = 0 in Ω ⊂ R^n with Neumann boundary conditions \frac{∂u}{∂v} = a(x)u under the rather mild structure conditions and without the concavity condition. We establish the global C^{1,Ω} estimates and the interior W^{2,p} estimates for W^{2,q}(Ω) solutions (q > 2n) by introducing new independent variables, and moreover prove the existence of W^{2,p}_{loc}(Ω)∩ C^{1,α}(\bar \Omega} viscosity solutions by using the accretive operator methods, where p E (0, 2), α ∈ (0, 1}.  相似文献   

12.
1.IntroductionThesolutionoftheC'-continuityrequlrementofKirchhoffbendingwithfiniteele-mentmodelsresult8incomplicatedhigher.l...nt.I2J'[4'I7].Besidesthelargenumberofunknowns,difficultiesmayalsoarisefrommiredsecondderiVativesattheverticestakenasnodalvari.bl.I8l.Toovercomesuchdifficulties,asplittingsplineelemelltmethodisintrod.cedl5j'l9],butthisalwayscausescomplicatedcomputation-nomthepracticalpoilitofviewlower-degreepolynomialfiniteelemelitsaremoredesirable.Unfortunately,thesimpleelementsbase…  相似文献   

13.
In this paper, we study the Cauchy problems u_t = Δu^m \quad u(x, 0) = μ and u_t = Δu^m - u^p \quad u(x, 0) = μ where p > 0, m > (1 - \frac{α}{n})^+ and μ is a finite Radon measure. We prove the uniqueness of solution and the existence of solution.  相似文献   

14.
This paper is concerned with the semilinear heat equation u_t = Δu - u^{-q} in Ω × (0, T) under the nonlinear boundary condition \frac{∂u}{∂v} = u^p on ∂Ω × (0, T). Criteria for finite time quenching and blow-up are established, quenching and blow-up sets are discussed, and the rates of quenching and blow-up are obtained.  相似文献   

15.
This paper considers a fast diffusion equation with potential ut= um V (x)um+upin Rn×(0,T), where 1 2αm+n< m ≤ 1, p > 1, n ≥ 2, V (x) ~ω|x|2with ω≥ 0 as |x| →∞,and α is the positive root of αm(αm + n 2) ω = 0. The critical Fujita exponent was determined as pc= m +2αm+nin a previous paper of the authors. In the present paper,we establish the second critical exponent to identify the global and non-global solutions in their co-existence parameter region p > pcvia the critical decay rates of the initial data.With u0(x) ~ |x| aas |x| →∞, it is shown that the second critical exponent a =2p m,independent of the potential parameter ω, is quite different from the situation for the critical exponent pc.  相似文献   

16.
Multiple blow-up for a porous medium equation with reaction   总被引:1,自引:0,他引:1  
The present paper is concerned with the Cauchy problem
$\left\{{ll}\partial_t u = \Delta u^m + u^p & \quad {\rm in}\; \mathbb R^N \times (0,\infty),\\ u(x,0) = u_0(x) \geq 0 & \quad {\rm in}\; \mathbb R^N, \right.$\left\{\begin{array}{ll}\partial_t u = \Delta u^m + u^p & \quad {\rm in}\; \mathbb R^N \times (0,\infty),\\ u(x,0) = u_0(x) \geq 0 & \quad {\rm in}\; \mathbb R^N, \end{array}\right.  相似文献   

17.
In this paper we generalize the method used to prove the Prime Number Theorem to deal with finite fields, and prove the following theorem:
$ \pi (x) = \frac{q} {{q - 1}}\frac{x} {{\log _q x}} + \frac{q} {{(q - 1)^2 }}\frac{x} {{\log _q^2 x}} + O\left( {\frac{x} {{\log _q^3 x}}} \right),x = q^n \to \infty $ \pi (x) = \frac{q} {{q - 1}}\frac{x} {{\log _q x}} + \frac{q} {{(q - 1)^2 }}\frac{x} {{\log _q^2 x}} + O\left( {\frac{x} {{\log _q^3 x}}} \right),x = q^n \to \infty   相似文献   

18.
The uniqueness of solution of the Cauchy problem u_t = Δu^m - u^p, \qquad S_T = R^n × (0, T) u(x, 0) = Φ(x), \qquad × ∈ R^n is obtained. Where n ≥ 1, m, p > 0, Φ(x) ∈ L^∞(R^n), Φ(x) ≥ 0.  相似文献   

19.
The connection between the functional inequalities
$f\left( {\frac{{x + y}} {2}} \right) \leqslant \frac{{f\left( x \right) + f\left( y \right)}} {2} + \alpha _J \left( {x - y} \right), x,y \in D,$f\left( {\frac{{x + y}} {2}} \right) \leqslant \frac{{f\left( x \right) + f\left( y \right)}} {2} + \alpha _J \left( {x - y} \right), x,y \in D,  相似文献   

20.
In this paper, we establish two families of approximations for the gamma function: $$ \begin{array}{lll} {\varGamma}(x+1)&=\sqrt{2\pi x}{\left({\frac{x+a}{{\mathrm{e}}}}\right)}^x {\left({\frac{x+a}{x-a}}\right)}^{-\frac{x}{2}+\frac{1}{4}} {\left({\frac{x+b}{x-b}}\right)}^{\sum\limits_{k=0}^m\frac{{\beta}_k}{x^{2k}}+O{{\left(\frac{1}{x^{2m+2}}\right)}}},\\ {\varGamma}(x+1)&=\sqrt{2\pi x}\cdot(x+a)^{\frac{x}{2}+\frac{1}{4}}(x-a)^{\frac{x}{2}-\frac{1}{4}} {\left({\frac{x-1}{x+1}}\right)}^{\frac{x^2}{2}}\\ &\quad\times {\left({\frac{x-c}{x+c}}\right)}^{\sum\limits_{k=0}^m\frac{{\gamma}_k}{x^{2k}}+O{\left({\frac{1}{x^{2m+2}}}\right)}}, \end{array}$$ where the constants ${\beta }_k$ and ${\gamma }_k$ can be determined by recurrences, and $a$ , $b$ , $c$ are parameters. Numerical comparison shows that our results are more accurate than Stieltjes, Luschny and Nemes’ formulae, which, to our knowledge, are better than other approximations in the literature.  相似文献   

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