Global attractors for the wave equation with nonlinear damping

Based on a new a priori estimate method, so-called asymptotic a priori estimate, the existence of a global attractor is proved for the wave equation utt+kg(ut)−Δu+f(u)=0 on a bounded domain Ω⊂R³ with Dirichlet boundary conditions. The nonlinear damping term g is supposed to satisfy the growth condition C₁(|s|−C₂)?|g(s)|?C₃(1+p|s|), where 1?p<5; the damping parameter is arbitrary; the nonlinear term f is supposed to satisfy the growth condition |f^′(s)|?C₄(1+q|s|), where q?2. It is remarkable that when 2<p<5, we positively answer an open problem in Chueshov and Lasiecka [I. Chueshov, I. Lasiecka, Long-time behavior of second evolution equations with nonlinear damping, Math. Scuola Norm. Sup. (2004)] and improve the corresponding results in Feireisl [E. Feireisl, Global attractors for damped wave equations with supercritical exponent, J. Differential Equations 116 (1995) 431-447].     

Existence of entire explosive positive radial solutions for a class of quasilinear elliptic systems

We show the existence of entire explosive positive radial solutions for quasilinear elliptic systems div(|∇u|^m−2∇u)=p(|x|)g(v), div(|∇v|ⁿ⁻²∇v)=q(|x|)f(u) on , where f and g are positive and non-decreasing functions on (0,∞) satisfying the Keller-Osserman condition.     

Isolated singularities of solutions to quasi-linear elliptic equations with absorption

We study the problem of removability of isolated singularities for a general second-order quasi-linear equation in divergence form −divA(x,u,∇u)+a₀(x,u)+g(x,u)=0 in a punctured domain Ω?{0}, where Ω is a domain in Rn, n?3. The model example is the equation −Δpu+gu|u|^p−2+u|u|^q−1=0, q>p−1>0, p<n. Assuming that the lower-order terms satisfy certain non-linear Kato-type conditions, we prove that for all point singularities of the above equation are removable, thus extending the seminal result of Brezis and Véron.     

Local existence of solutions to some degenerate parabolic equation associated with the p-Laplacian

The existence of local (in time) solutions of the initial-boundary value problem for the following degenerate parabolic equation: ut(x,t)−Δpu(x,t)−|u|^q−2u(x,t)=f(x,t), (x,t)∈Ω×(0,T), where 2?p<q<+∞, Ω is a bounded domain in RN, is given and Δp denotes the so-called p-Laplacian defined by Δpu:=∇⋅(|∇u|^p−2∇u), with initial data u₀∈Lr(Ω) is proved under r>N(q−p)/p without imposing any smallness on u₀ and f. To this end, the above problem is reduced into the Cauchy problem for an evolution equation governed by the difference of two subdifferential operators in a reflexive Banach space, and the theory of subdifferential operators and potential well method are employed to establish energy estimates. Particularly, Lr-estimates of solutions play a crucial role to construct a time-local solution and reveal the dependence of the time interval [0,T₀] in which the problem admits a solution. More precisely, T₀ depends only on Lr|u₀| and f.     

Nontrivial solutions for perturbations of a Hardy-Sobolev operator on unbounded domains

In this paper we study the existence of nontrivial solution of the problem −Δ_pu−(μ/[d(x)]^p)|u|^p−2u=f(u) in Ω and u=0 on ∂Ω, where is a bounded domain with smooth boundary in Existence is established using mountain-pass lemma and concentration of compactness principle.     

On the Cauchy problem for the evolution p-Laplacian equations with gradient term and source

In this paper, the authors study the equation ut=div(|Du|^p−2Du)+|u|^q−1u−λl|Du| in RN with p>2. We first prove that for 1?l?p−1, the solution exists at least for a short time; then for , the existence and nonexistence of global (in time) solutions are studied in various situations.     

Linking solutions for p-Laplace equations with nonlinearity at critical growth

Under a suitable condition on n and p, the quasilinear equation at critical growth −Δpu=λ|u|^p−2u+|u|^p∗−2u is shown to admit a nontrivial weak solution for any λ?λ₁. Nonstandard linking structures, for the associated functional, are recognized.     

Ground state solutions for a semilinear elliptic problem with a convection term

By a sub-supersolution method and a perturbed argument, we improve the earlier results concerning the existence of ground state solutions to a semilinear elliptic problem −Δu+p(x)q|∇u|=f(x,u), u>0, x∈RN, , where q∈(1,2], for some α∈(0,1), p(x)?0, ∀x∈RN, and f:RN×(0,∞)→[0,∞) is a locally Hölder continuous function which may be singular at zero.     

Existence of a solution to a nonlinear equation

Equation (−Δ+k²)u+f(u)=0 in D, u_|∂D=0, where k=const>0 and D⊂R³ is a bounded domain, has a solution if is a continuous function in the region |u|?a, piecewise-continuous in the region |u|?a, with finitely many discontinuity points uj such that f(uj±0) exist, and uf(y)?0 for |u|?a, where a?0 is an arbitrary fixed number.     

Nonresonance between the first two eigenvalues for a Steklov problem

In this paper, we study the solvability of the Steklov problem Δ_pu=|u|^p−2u in Ω, on ∂Ω, under assumptions on the asymptotic behaviour of the quotients f(x,s)/|s|^p−2s and pF(x,s)/|s|^p which extends the classical results with Dirichlet boundary conditions that for a.e. x∈∂Ω, the limits at the infinity of these quotients lie between the first two eigenvalues.     

Positive solutions for some Schrödinger equations having partially periodic potentials

This paper is concerned with the problem of finding positive solutions of the equation −Δu+(a_∞+a(x))u=|u|^q−2u, where q is subcritical, Ω is either RN or an unbounded domain which is periodic in the first p coordinates and whose complement is contained in a cylinder , a_∞>0, a∈C(RN,R) is periodic in the first p coordinates, infx∈RN(a_∞+a(x))>0 and a(x^′,x^″)→0 as |x^″|→∞ uniformly in x^′. The cases a?0 and a?0 are considered and it is shown that, under appropriate assumptions on a, the problem has one solution in the first case and p+1 solutions in the second case when p?N−2.     

Distance-two labellings of Hamming graphs

Let j≥k≥0 be integers. An ?-L(j,k)-labelling of a graph G=(V,E) is a mapping ?:V→{0,1,2,…,?} such that |?(u)−?(v)|≥j if u,v are adjacent and |?(u)−?(v)|≥k if they are distance two apart. Let λ_j,k(G) be the smallest integer ? such that G admits an ?-L(j,k)-labelling. Define to be the smallest ? if G admits an ?-L(j,k)-labelling with ?(V)={0,1,2,…,?} and ∞ otherwise. An ?-cyclic L(j,k)-labelling is a mapping ?:V→Z_? such that |?(u)−?(v)|_?≥j if u,v are adjacent and |?(u)−?(v)|_?≥k if they are distance two apart, where |x|_?=min{x,?−x} for x between 0 and ?. Let σ_j,k(G) be the smallest ?−1 of such a labelling, and define similarly to . We determine λ_2,0, , σ_2,0 and for all Hamming graphs K_q1□K_q2□?□K_qd (d≥2, q₁≥q₂≥?≥q_d≥2) and give optimal labellings, with the only exception being for q≥4. We also prove the following “sandwich theorem”: If q₁ is sufficiently large then for any graph G between K_q1□K_q2 and K_q1□K_q2□?□K_qd, and moreover we give a labelling which is optimal for these eight invariants simultaneously.     

Existence and multiplicity of solutions for quasilinear nonhomogeneous problems: An Orlicz-Sobolev space setting

We study the boundary value problem −div(log(1+q|∇u|)|∇u|^p−2∇u)=f(u) in Ω, u=0 on ∂Ω, where Ω is a bounded domain in RN with smooth boundary. We distinguish the cases where either f(u)=−λ|u|^p−2u+|u|^r−2u or f(u)=λ|u|^p−2u−|u|^r−2u, with p, q>1, p+q<min{N,r}, and r<(Np−N+p)/(N−p). In the first case we show the existence of infinitely many weak solutions for any λ>0. In the second case we prove the existence of a nontrivial weak solution if λ is sufficiently large. Our approach relies on adequate variational methods in Orlicz-Sobolev spaces.     

Self-similar singular solutions of a p-Laplacian evolution equation with absorption

We consider, for p∈(1,2) and q>1, self-similar singular solutions of the equation v_t=div(|∇v|^p−2∇v)−v^q in Rn×(0,∞); here by self-similar we mean that v takes the form v(x,t)=t^−αw(|x|t^−αβ) for α=1/(q−1) and β=(q+1−p)/p, whereas singular means that v is non-negative, non-trivial, and for all x≠0. That is, we consider the ODE problem

(0.1)     

Mathematical analysis to a nonlinear fourth-order partial differential equation

The paper first study the steady-state thin film type equation

∇⋅(uⁿ|∇Δu|^q−2∇Δu)−δu^mΔu=f(x,u)     

Stable transition layers in a semilinear diffusion equation with spatial inhomogeneities in N-dimensional domains

We prove existence and establish the asymptotic behavior, as ε→0, of stable stationary solutions to the equation u_t=ε∇·[d(x)∇u]+(1−u²)[u−a(x)], for , where , N?2, with Neumann boundary condition. The function a(x)∈C0,ν(Ω) satisfies −1<a(x)<1 and vanishes on some hypersurfaces. The results generalize to N-dimensional domains and to variable diffusivity earlier paper by Angenent et al. (J. Differential Equations 67 (1987) 212).