Pullback attractors for a class of non-autonomous nonclassical diffusion equations

We consider the first initial boundary value problem for the non-autonomous nonclassical diffusion equation u_t−εΔu_t−Δu+f(u)=g(t), ε∈[0,1], in a bounded domain in R^N. Under a Sobolev growth rate of the nonlinearity f and a suitable exponential growth of the external force g, using the asymptotic a priori estimate method, we prove the existence of pullback D-attractors in the space and the upper semicontinuity of at ε=0.     

Large solutions to non-monotone semilinear elliptic systems

We present the existence of entire large positive radial solutions for the non-monotonic system Δu=p(|x|)g(v), Δv=q(|x|)f(u) on Rn where n?3. The functions f and g satisfy a Keller-Osserman type condition while nonnegative functions p and q are required to satisfy the decay conditions and . Further, p and q are such that min(p,q) does not have compact support.     

Hyperbolic systems of balance laws via vanishing viscosity

Global weak solutions of a strictly hyperbolic system of balance laws in one-space dimension are constructed by the vanishing viscosity method of Bianchini and Bressan. For global existence, a suitable dissipativeness assumption has to be made on the production term g. Under this hypothesis, the viscous approximations u^?, that are globally defined solutions to , satisfy uniform BV bounds exponentially decaying in time. Furthermore, they are stable in L¹ with respect to the initial data. Finally, as ?→0, u^? converges in to the admissible weak solution u of the system of balance laws u_t+(f(u))_x+g(u)=0 when A=Df.     

On the existence of local strong solutions for the Navier-Stokes equations in completely general domains

There are only very few results on the existence of unique local in time strong solutions of the Navier-Stokes equations for completely general domains Ω⊆R³, although domains with edges and corners, bounded or unbounded, are very important in applications. The reason is that the L^q-theory for the Stokes operator A is available in general only in the Hilbert space setting, i.e., with q=2. Our main result for a general domain Ω is optimal in a certain sense: Consider an initial value and a zero external force. Then the condition is sufficient and necessary for the existence of a unique local strong solution u∈L⁸(0,T;L⁴(Ω)) in some interval [0,T), 0<T≤∞, with u(0)=u₀, satisfying Serrin’s condition . Note that Fujita-Kato’s sufficient condition u₀∈D(A^1/4) is strictly stronger and therefore not optimal.     

The Brezis-Nirenberg problem near criticality in dimension 3

We consider the problem of finding positive solutions of Δu+λu+u^q=0 in a bounded, smooth domain Ω in , under zero Dirichlet boundary conditions. Here q is a number close to the critical exponent 5 and 0<λ<λ₁. We analyze the role of Green's function of Δ+λ in the presence of solutions exhibiting single and multiple bubbling behavior at one point of the domain when either q or λ are regarded as parameters. As a special case of our results, we find that if , where λ∗ is the Brezis-Nirenberg number, i.e., the smallest value of λ for which least energy solutions for q=5 exist, then this problem is solvable if q>5 and q−5 is sufficiently small.     

Regularity criterion of axisymmetric weak solutions to the 3D Navier-Stokes equations

We consider the regularity of axisymmetric weak solutions to the Navier-Stokes equations in R³. Let u be an axisymmetric weak solution in R³×(0,T), w=curlu, and wθ be the azimuthal component of w in the cylindrical coordinates. Chae-Lee [D. Chae, J. Lee, On the regularity of axisymmetric solutions of the Navier-Stokes equations, Math. Z. 239 (2002) 645-671] proved the regularity of weak solutions under the condition wθ∈Lq(0,T;Lr), with , . We deal with the marginal case r=∞ which they excluded. It is proved that u becomes a regular solution if .     

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.     

Entire large solutions to semilinear elliptic systems

We consider the problem of existence of positive solutions to the elliptic system Δu=p(|x|)vα, Δv=q(|x|)uβ on Rn (n?3) which satisfies . The parameters α and β are positive, and the nonnegative functions p and q are continuous and min{p(r),q(r)} does not have compact support. We show that if αβ?1, then such a solution exists if and only if the functions p and q satisfy

     

Expansions in non-integer bases: Lower, middle and top orders

Let q∈(1,2); it is known that each x∈[0,1/(q−1)] has an expansion of the form with an∈{0,1}. It was shown in [P. Erd?s, I. Joó, V. Komornik, Characterization of the unique expansions and related problems, Bull. Soc. Math. France 118 (1990) 377-390] that if , then each x∈(0,1/(q−1)) has a continuum of such expansions; however, if , then there exist infinitely many x having a unique expansion [P. Glendinning, N. Sidorov, Unique representations of real numbers in non-integer bases, Math. Res. Lett. 8 (2001) 535-543]. In the present paper we begin the study of parameters q for which there exists x having a fixed finite number m>1 of expansions in base q. In particular, we show that if q<q₂=1.71…, then each x has either 1 or infinitely many expansions, i.e., there are no such q in . On the other hand, for each m>1 there exists γm>0 such that for any q∈(2−γm,2), there exists x which has exactly m expansions in base q.     

An elliptic system with bifurcation parameters on the boundary conditions

In this paper we consider the elliptic system Δu=a(x)upvq, Δv=b(x)urvs in Ω, a smooth bounded domain, with boundary conditions , on ∂Ω. Here λ and μ are regarded as parameters and p,s>1, q,r>0 verify (p−1)(s−1)>qr. We consider the case where a(x)?0 in Ω and a(x) is allowed to vanish in an interior subdomain Ω₀, while b(x)>0 in . Our main results include existence of nonnegative nontrivial solutions in the range 0<λ<λ₁?∞, μ>0, where λ₁ is characterized by means of an eigenvalue problem, and the uniqueness of such solutions. We also study their asymptotic behavior in all possible cases: as both λ,μ→0, as λ→λ₁<∞ for fixed μ (respectively μ→∞ for fixed λ) and when both λ,μ→∞ in case λ₁=∞.     

On the dynamics of a class of nonclassical parabolic equations

We consider the first initial and boundary value problem of nonclassical parabolic equations ut−μΔut−Δu+g(u)=f(x) on a bounded domain Ω, where μ∈[0,1]. First, we establish some uniform decay estimates for the solutions of the problem which are independent of the parameter μ. Then we prove the continuity of solutions as μ→0. Finally we show that the problem has a unique global attractor Aμ in in the topology of H²(Ω); moreover, Aμ→A₀ in the sense of Hausdorff semidistance in as μ goes to 0.     

On global attractor for m-Laplacian parabolic equation with local and nonlocal nonlinearity

In this paper, we study the long-time behavior of solutions for m-Laplacian parabolic equation in Ω×(0,∞) with the initial data u(x,0)=u₀(x)∈Lq, q?1, and zero boundary condition in ∂Ω. Two cases for a(x)?a₀>0 and a(x)?0 are considered. We obtain the existence and Lp estimate of global attractor A in Lp, for any p?max{1,q}. The attractor A is in fact a bounded set in if a(x)?a₀>0 in Ω, and A is bounded in if a(x)?0 in Ω.     

A free boundary problem for p-Laplacian in the plane

We consider the following free boundary problem in an unbounded domain Ω in two dimensions: Δpu=0 in Ω, on J₀, on J₁, where ∂Ω=J₀∪J₁. We prove that if 0<u<1 in Ω, Ji is the graph of a function in and gi is a constant for each i=0,1, then the free boundary ∂Ω must be two parallel straight lines and the solution u must be a linear function. The proof is based on maximum principle.     

The asymptotic behaviour of the unique solution for the singular Lane-Emden-Fowler equation

By Karamata regular variation theory and constructing comparison functions, we show the exact asymptotic behaviour of the unique classical solution near the boundary to a singular Dirichlet problem −Δu=k(x)g(u), u>0, x∈Ω, u_|∂Ω=0, where Ω is a bounded domain with smooth boundary in RN; g∈C¹((0,∞),(0,∞)), , for each ξ>0, for some γ>0; and for some α∈(0,1), is nonnegative on Ω, which is also singular near the boundary.     

The heat equation with singular nonlinearity and singular initial data

We study the existence, uniqueness and regularity of positive solutions of the parabolic equation ut−Δu=a(x)uq+b(x)up in a bounded domain and with Dirichlet's condition on the boundary. We consider here a∈Lα(Ω), b∈Lβ(Ω) and 0<q?1<p. The initial data u(0)=u₀ is considered in the space Lr(Ω), r?1. In the main result (0<q<1), we assume a,b?0 a.e. in Ω and we assume that u₀?γdΩ for some γ>0. We find a unique solution in the space .     

On positive solutions for some semilinear periodic parabolic eigenvalue problems

For a bounded domain Ω in , N?2, satisfying a weak regularity condition, we study existence of positive and T-periodic weak solutions for the periodic parabolic problem Lu_λ=λg(x,t,u_λ) in , u_λ=0 on . We characterize the set of positive eigenvalues with positive eigenfunctions associated, under the assumptions that g is a Caratheodory function such that ξ→g(x,t,ξ)/ξ is nonincreasing in (0,∞) a.e. satisfying some integrability conditions in (x,t) and