Singular boundary value problems for the one-dimension p-Laplacian

The singular boundary value problem

where φ(s)=|s|^p−2s, p>1, is studied in this paper. The singularity may appear at u=0, t=0 and t=1, and the function g may change sign. The existence of solutions is obtained via an upper and lower solution method.     

A two phase elliptic singular perturbation problem with a forcing term

We study the following two phase elliptic singular perturbation problem:

Δu^ε=β_ε(u^ε)+f^ε,

in , where ε>0, , with β a Lipschitz function satisfying β>0 in (0,1), β≡0 outside (0,1) and . The functions u^ε and f^ε are uniformly bounded. One of the motivations for the study of this problem is that it appears in the analysis of the propagation of flames in the high activation energy limit, when sources are present.We obtain uniform estimates, we pass to the limit (ε→0) and we show that limit functions are solutions to the two phase free boundary problem:

where f=limf^ε, in a viscosity sense and in a pointwise sense at regular free boundary points.In addition, we show that the free boundary is smooth and thus limit functions are classical solutions to the free boundary problem, under suitable assumptions.Some of the results obtained are new even in the case f^ε≡0.The results in this paper also apply to other combustion models. For instance, models with nonlocal diffusion and/or transport. Several of these applications are discussed here and we get, in some cases, the full regularity of the free boundary.     

A Bahri–Lions theorem revisited

In 1988, A. Bahri and P.L. Lions [A. Bahri, P.L. Lions, Morse-index of some min–max critical points. I. Application to multiplicity results, Comm. Pure Appl. Math. 41 (1988) 1027–1037] studied the following elliptic problem:

where Ω is a bounded smooth domain of , 2<p<(2N−2)/(N−2) and f(x,u) is not assumed to be odd in u. They proved the existence of infinitely many solutions under an appropriate growth restriction on f. In the present paper, we improve this result by showing that under the same growth assumption on f the problem admits in fact infinitely many sign-changing solutions. In addition we derive an estimate on the number of their nodal domains. We also deal with the corresponding fourth order equation Δ²u=|u|^p−2u+f(x,u) with both Dirichlet and Navier boundary conditions, as well as with strongly coupled elliptic systems.     

Long time behaviour for generalized complex Ginzburg–Landau equation

In this paper, the two-dimensional generalized complex Ginzburg–Landau equation (CGL)

u_t=ρu−Δφ(u)−(1+iγ)Δu−νΔ²u−(1+iμ)|u|^2σu+αλ₁(|u|²u)+β(λ₂)|u|²

is studied. The existence of global attractor for this equation with periodic boundary condition is established and upper bounds of Hausdorff and fractal dimensions of attractor are obtained.     

Existence of multiple positive solutions for nonlinear m-point boundary-value problems

In this paper, we afford some sufficient conditions to guarantee the existence of multiple positive solutions for the nonlinear m-point boundary-value problem for the one-dimensional p-Laplacian

(φ_p(u^′))^′+f(t,u)=0, t(0,1),

     

Marcinkiewicz–Zygmund strong laws for -statistics of weakly dependent observations

We prove the Marcinkiewicz–Zygmund Strong Law of Large Numbers for U-statistics of strictly stationary, absolutely regular observations (ξ_i)_i≥1. Under suitable moment conditions and conditions on the mixing rate, we show that

for some γ≥0, in the non-degenerate case, and

in the degenerate case.     

Global existence and blow-up of solutions for a system of nonlinear viscoelastic wave equations with damping and source

In this paper we investigate the global existence and finite time blow-up of solutions to the system of nonlinear viscoelastic wave equations

in Ω×(0,T) with initial and Dirichlet boundary conditions, where Ω is a bounded domain in . Under suitable assumptions on the functions g_i(), , the initial data and the parameters in the equations, we establish several results concerning local existence, global existence, uniqueness and finite time blow-up property.     

Positive solutions for one-dimensional -Laplacian boundary value problems with sign changing nonlinearity

This paper deals with the existence of positive solutions for the one-dimensional p-Laplacian

subject to the boundary value conditions:

where _p(s)=|s|^p−2s,p>1. We show that it has at least one or two positive solutions under some assumptions by applying the fixed point theorem. The interesting points are that the nonlinear term f is involved with the first-order derivative explicitly and f may change sign.     

Positive solutions for a class of p-Laplace problems involving quasi-linear and semi-linear terms

The aim of this paper is to discuss the positive solutions of the p-Laplace problem

−div(|u|^p−2u)+g(u)|u|^p=λu^q,

where p>1, q>1, is a nonnegative continuous function, λ is a real number. The sufficient condition to have positive solutions of the above problem is . However, if , there is no solution which belongs to it. Therefore, our results are optimal.     

Curves of positive solutions of boundary value problems on time-scales

Let TR be a time-scale, with a=infT, b=supT. We consider the nonlinear boundary value problem

(2)

(4)

u(a)=u(b)=0,

where λR₊:=[0,∞), and satisfies the conditions

We prove a strong maximum principle for the linear operator defined by the left-hand side of (1), and use this to show that for every solution (λ,u) of (1)–(2), u is positive on T a,b . In addition, we show that there exists λ_max>0 (possibly λ_max=∞), such that, if 0λ<λ_max then (1)–(2) has a unique solution u(λ), while if λλ_max then (1)–(2) has no solution. The value of λ_max is characterised as the principal eigenvalue of an associated weighted eigenvalue problem (in this regard, we prove a general existence result for such eigenvalues for problems with general, nonnegative weights).     

THE EXISTENCE OF GLOBAL SOLUTION AND THE BLOWUP PROBLEM FOR SOME p-LAPLACE HEAT EQUATIONS

This article consider, for the following heat equation ut/|x|s-△pu=uq,(x,t)∈Ω×(0,T), u(x,t)=0,(x,t)∈(?)Ω×(0,T), u(x,0)=u0(x),u0(x)≥0,u0(x)(?)0 the existence of global solution under some conditions and give two sufficient conditions for the blow up of local solution in finite time, whereΩis a smooth bounded domain in RN(N>p),0∈Ω,△pu=div(|▽u|p-2▽u),0≤s≤2,p≥2,p-1相似文献   

Almost periodic and almost automorphic solutions to semilinear parabolic boundary differential equations

In this paper, we study the existence and uniqueness of almost periodic and almost automorphic solutions to the semilinear parabolic boundary differential equations

where AA_m|kerL generates a hyperbolic analytic semigroup on a Banach space X. The functions h and are defined on some intermediate subspaces X_β,0<β<1, and take values in X and in a boundary space ∂X respectively.     

Blow-up analysis for a system of heat equations with nonlinear flux which obey different laws

We consider a system of heat equations u_t=Δu and v_t=Δv in Ω×(0,T) completely coupled by nonlinear boundary conditions

We prove that the solutions always blow up in finite time for non-zero and non-negative initial values. Also, the blow-up only occurs on ∂Ω with

for p,q>0, 0≤α<1 and 0≤β<p.     

BV solutions to a degenerate parabolic equation for image denoising

In this article, the authors consider equation ut = div(ψ(Гu)A(|Du|2)Du) -(u- I), where ψ is strictly positive and Г is a known vector-valued mapping, A: R → R is decreasing and A(s) ～ 1/√a as s → ∞. This kind of equation arises naturally from image denoising. For an initial datum I ∈ BVloc ∩ L∞, the existence of BV solutions to the initial value problem of the equation is obtained.     

Hardy–Sobolev critical elliptic equations with boundary singularities

Unlike the non-singular case s=0, or the case when 0 belongs to the interior of a domain Ω in (n3), we show that the value and the attainability of the best Hardy–Sobolev constant on a smooth domain Ω,

when 0<s<2, , and when 0 is on the boundary ∂Ω are closely related to the properties of the curvature of ∂Ω at 0. These conditions on the curvature are also relevant to the study of elliptic partial differential equations with singular potentials of the form:

where f is a lower order perturbative term at infinity and f(x,0)=0. We show that the positivity of the sectional curvature at 0 is relevant when dealing with Dirichlet boundary conditions, while the Neumann problems seem to require the positivity of the mean curvature at 0.     

A note on asymptotic behavior for negative drift random walk with dependent heavy-tailed steps and its application to risk theory

In this article, the dependent steps of a negative drift random walk are modelled as a two-sided linear process Xn =-μ ∞∑j=-∞ψn-jεj, where {ε, εn; -∞＜ n ＜ ∞}is a sequence of independent, identically distributed random variables with zero mean, μ＞0 is a constant and the coefficients {ψi;-∞＜ i ＜∞} satisfy 0 ＜∞∑j=-∞|jψj| ＜∞. Under the conditions that the distribution function of |ε| has dominated variation and ε satisfies certain tail balance conditions, the asymptotic behavior of P{supn≥0(-nμ ∞∑j=-∞εjβnj) ＞ x}is discussed. Then the result is applied to ultimate ruin probability.     

RELATIONS BETWEEN PACKING PREMEASURE AND MEASURE ON METRIC SPACE

Let X be a metric space andμa finite Borel measure on X. Let pμq,t and pμq,t be the packing premeasure and the packing measure on X, respectively, defined by the gauge (μB(x,r))q(2r)t, where q, t∈R. For any compact set E of finite packing premeasure the authors prove: (1) if q≤0 then pμq,t(E)=pμq,t(E);(2)if q>0 andμis doubling on E then pμq,t(E) and pμq,t(E) are both zero or neither.     

Multiplicity of positive solutions for a class of quasilinear nonhomogeneous Neumann problems

In this paper we study the existence, nonexistence and multiplicity of positive solutions for nonhomogeneous Neumann boundary value problem of the type

where Ω is a bounded domain in with smooth boundary, 1<p<n,Δ_pu=div(|u|^p-2u) is the p-Laplacian operator, , , (x)0 and λ is a real parameter. The proofs of our main results rely on different methods: lower and upper solutions and variational approach.     

Nonradial large solutions of sublinear elliptic problems

Let p be a nonnegative locally bounded function on , N3, and 0<γ<1. Assuming that the oscillation sup_|x|=rp(x)−inf_|x|=rp(x) tends to zero as r→∞ at a specified rate, it is shown that the equation Δu=p(x)u^γ admits a positive solution in satisfying lim_|x|→∞u(x)=∞ if and only if

     

Stability and instability of periodic standing wave solutions for some Klein–Gordon equations

In the present paper we show some results concerning the orbital stability of dnoidal standing wave solutions and orbital instability of cnoidal standing wave solutions to the following Klein–Gordon equation:

u_tt−u_xx+u−|u|²u=0.