Regularity Results for a Strongly Degenerate Parabolic Equation

M. Bertsch & R. Dal Passo proved the existence and uniqueness of the Cauchy problem for u_t = (φ(u),ψ(u_x))_x, where φ > 0, ψ is a strictly increasing function with lim_{s → ∞}ψ(s) = ψ_∞ < ∞. The regularity of the solution has been obtained under the condition φ" < 0 or φ = const. In the present paper, under the condition φ" ≤ 0, we give some regularity results. We show that the solution can be classical after a finite time. Further, under the condition φ" ≤ -α_0 (where -α_0 is a constant), we prove the gradient of the solution converges to zero uniformly with respect to x as t → +∞.     

On the Cauchy Problem and Initial Trace for Nonlinear Filtration Type with Singularity

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.     

Stability of the solutions of the Cauchy problem for first-order quasilinear equations

In this paper we establish estimates of the stability of the solutions of the Cauchy problem for the multidimensional quasilinear equation \(u_t + (\varphi _i (u))_{x_i } = \psi (t, x, u)\) relative to variation of the function ?_i(u), ψ(t, x, u).     

Cauchy Problem for Semilinear Wave Equations in Four Space Dimensions with Small Initial Data

In this paper, we consider the Cauchy problem ◻u(t,x) = |u(t,x)|^p, (t,x) ∈ R^+ × R^4 t = 0 : u = φ(x), u_t = ψ(x), x ∈ R^4 where ◻ = ∂²_t - Σ^4_{i=1}∂²_x_i, is the wave operator, φ, ψ ∈ C^∞_0 (R^4). We prove that for p > 2 the problem has a global solution provided tile initial data is sufficiently small.     

Global W2,p Solutions of GBBM Equations on Unbounded Domain

In this paper we study the initial boundary value problem of GBBM equations on unbounded domain u_t - Δu_t = div f(u) u(x,0) = u_0(x) u|_{∂Ω} = 0 and corresponding Cauchy problem. Under the conditions: f( s) ∈ C^sup1 and satisfies (H)\qquad |f'(s)| ≤ C|s|^ϒ, 0 ≤ ϒ ≤ \frac{2}{n-2} if n ≥ 3; 0 ≤ ϒ < ∞ if n = 2 u_0(x) ∈ W^{2,p}(Ω) ∩ W^{2,2}(Ω) ∩ W^{1,p}_0(Ω)(W^{2,p}(R^n) ∩ W^{2,2}(R^n) for Cauchy problem), 2 ≤ p < ∞, we obtain the existence and uniqueness of global solution u(x, t) ∈ W^{1,∞}(0, T; W^{2,p}(Ω) ∩ W^{2,2}(Ω) ∩ W^{1,p}_0(Ω))(W^{1,∞}(0, T; W^{2,p}(R^n) ∩ W^{2,2} (R^n)) for Cauchy problem), so the results of [1] and [2] are generalized and improved in essential.     

Asymptotic behavior relative to a large parameter of the solution of the Fok-Klein-Gordon equation in the case of a discontinuous initial condition

One considers the problem of the asymptotic behavior for K→+∞ of the solution of the Cauchy problem $$u_{tt} - u_{xx} + \kappa ^2 u = 0; u|_{t = 0} = \theta (x), u_t |_{t = 0} = 0 (t > 0 - fixed)$$ Hereθ(x) is the Heaviside function. In the neighborhood of the characteristics x=±t function u(x,t)_?2 oscillates exceptionally fast (the wavelength is of order k^?2). Near the t axis the asymptotics of u(x,t) contains the Fresnel integral.     

Existence and asymptotic stability of a viscoelastic wave equation with a delay

In this paper, we consider the viscoelastic wave equation with a delay term in internal feedbacks; namely, we investigate the following problem

Hyponormality of Toeplitz Operators with Polynomial Symbols: An Extremal Case

If T_φ is a hyponormal Toeplitz operator with polynomial symbol φ = ḡ + f (f, g ∈ H^∞ (𝕋 )) such that g divides f, and if ψ := then where μ is the leading coefficient of ψ and 𝒵(ψ) denotes the set of zeros of ψ. In this paper we present a necessary and sufficient condition for T_φ to be hyponormal when φ enjoys an extremal case in the above inequality, that is, equality holds in the above inequality.     

THE GLOBAL SMOOTH SOLUTIONS OF SECOND ORDER QUASILINEAR HYPERBOLIC EQUATIONS WITH DISSIPATIVE BOUNDARY CONDITIONS

The paper deals with the following boundary problem of the second order quasilinear hyperbolic equation with a dissipative boundary condition on a part of the boundary:u_(tt)-sum from i,j=1 to n a_(ij)(Du)u_(x_ix_j)=0, in (0, ∞)×Ω,u|Γ_0=0,sum from i,j=1 to n, a_(ij)(Du)n_ju_x_i+b(Du)u_t|Γ_1=0,u|t=0=φ(x), u_t|t=0=ψ(x), in Ω, where Ω=Γ_0∪Γ_1, b(Du)≥b_0>0. Under some assumptions on the equation and domain, the author proves that there exists a global smooth solution for above problem with small data.     

Global W2,p (2<=p

This paper studies the initial-boundary value problem of GBBM equations u_t - Δu_t = div f(u) \qquad\qquad\qquad(a) u(x, 0) = u_0(x)\qquad\qquad\qquad(b) u |∂Ω = 0 \qquad\qquad\qquad(c) in arbitrary dimensions, Ω ⊂ R^n. Suppose that. f(s) ∈ C¹ and |f'(s)| ≤ C (1+|s|^ϒ), 0 ≤ ϒ ≤ \frac{2}{n-2} if n ≥ 3, 0 ≤ ϒ < ∞ if n = 2, u_0 (x) ∈ W^{2⋅p}(Ω) ∩ W^{1⋅p}_0(Ω) (2 ≤ p < ∞), then ∀T > 0 there exists a unique global W^{2⋅p} solution u ∈ W^{1,∞}(0, T; W{2⋅p}(Ω)∩ W^{1⋅p}_0(Ω)), so the known results are generalized and improved essentially.     

Products of Distributions, Conservation Laws and the Propagation of δ -Shock Waves

This paper contains a study of propagation of singular travelling waves u(x,t)for conservation laws ut + [φ(u)]x =Ψ(u),where φ,Ψ are entire functions taking real values on the real axis.Conditions for ...     

一类非线性奇异微分方程正解的存在性定理

设(i) f(t,u): (0,1)×(0,+∞)→[0,+∞)连续，关于u 单调增加; (ii) 存在函数g:[1,+∞)→(0,+∞),g(b)0,G(t,s)是相应问题的Green函数。     

Blow-up of Solutions of a Quasilinear Wave Equation with Nonlinear Dissipation

In this paper we establish the blow up of solutions to the quasilinear wave equation with a nonlinear dissipative term u_{tt} - M(||A^{1/2}u||²_2)Au + |ut|^βu_t = |u|^pu x ∈ Ω, t > 0     

A New Viscous Regularization of the Riemann Problem for Burgers' Equation

This paper gives a new viscous regularization of the Riemann problem for Burgers' equation u_t + (\frac{u²}{2})_z = 0 with Riemann initial data u = u_(x ≤ 0), u = u_+(x > 0} at t = 0. The regularization is given by u_t + (\frac{u²}{2})_z = εe^tu_{zz} with appropriate initial data. The method is different from the classical method, through comparison of three viscous equations of it. Here it is also shown that the difference of the three regularizations approaches zero in appropriate integral norms depending on the data as ε → 0_+ for any given T > 0.     

DOUBLE Φ-FUNCTION INEQUALITY FOR NONNEGATIVE SUBMARTINGALES

91.IntroductionLet4beanonnegativenondecreasingcontinuousfunctionon[0,co)with.(0)=0andi454(t)=co,and(fi,Z,p)beacompleteprobabilityspace.WedenotebyMthesetofallZ-measurablefunctionsandL'(fl)={fEM,3e>0,E.(elfl)相似文献   

一维p-Laplacian方程正解的存在性

本文考虑一维p-Laplacian非线性边值问题(ψ(u′))′+f(t,u)=0,αψ(u(0))—βψ(u′(0))=O,γψ(u(1))+δψ(u′(1))=0,其中ψ(s):=|s|~(p-2)s,P>1.通过应用Krasnoselskii锥不动点定理,建立了该问题存在多个正解的充分条件,推广并丰富了以往文献的一些结论.     

集值映射的不动点指数与带间断非线性项的椭圆型方程的多重解

<正> 本文研究二阶半线性椭圆边值问题■的多重解(符号详见§3),其中φ(x,t)允许对t是不连续的.一些自由边界问题可以化归这类问题.为了统一处理φ(x,t)对t连续与不连续两种情形,我们采用集值映射的观点.为此推广了经典的算子与Hammerstein算子到集值映射,并发展了集值映射的Leray-Schauder度理论;与已有的集值映射理论不同,现在处理的是映射串(定     

一类三阶拟线性微分方程非振动解的渐近性

主要讨论的是一类三阶拟线性微分方程(p(t)|u″|~(α-1)u″)′+q(t)|u|~(β-1)u=0其中α0,β0,p(t)和q(t)是定义在区间[a,∞)上的连续函数,且满足当t≥a时p(t)0,q(t)0.当t→∞时此方程满足∫_a~∞1/((p(t))~(1/α))dt=∞的特殊非振动解存在的充分必要条件.     

Blow-up of the solution for higher-order Kirchhoff-type equations with nonlinear dissipation

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.     

Generalized Solution of the First Boundary Value Problem for Parabolic Monge-Ampere Equation

The existence and uniqueness of generalized solution to the first boundary value problem for parabolic Monge-Ampère equation - ut det D²_xu = f in Q = Ω × (0, T], u = φ on ∂_pQ are proved if there exists a strict generalized supersolution u_φ, where Ω ⊂ R^n is a bounded convex set, f is a nonnegative bounded measurable function defined on Q, φ ∈ C(∂_pQ), φ(x, 0) is a convex function in \overline{\Omega}, ∀x_0 ∈ ∂Ω, φ(x_0, t) ∈ C^α([0, T]).