A note on the problem −Δu=λu+u|u|2*−2

The dual approach to the problem –u=u+u|u|^2*–2, uþe ₀ ¹ (gW), permits a simple proof of a recent existence result [5] and allows extensions of this result to similar problems also with asymmetric mnonlinearities.Supported by min P.I., Gruppo Naz. (40%) Calcolo delle Variazioni...     

On radially symmetric solutions of the equation ?Δu + u = |u| p-1 u. An ODE approach

Questions of the existence in a ball of radially symmetric solutions of the equation indicated in the title with the Dirichlet zero boundary conditions are studied in many publications and generally speaking, there was obtained more or less complete answer on these questions. It is known now that if the dimension of the space d????3 and 1 <?p?<?(d?+ 2)/(d ? 2) or if d?=?2 and p?>?1, then for any integer l??? 0 this problem in a ball or in the entire space ${x \in \mathbb {R}^d}$ has a radially symmetric solution with precisely l zeros as a function of r?=?|x|. If d??? 3 and p????(d?+?2)/(d ? 2), then the problem in the entire space has no nontrivial solution. For the first time, this problem was studied by a variant of the variational method. However, it is known to the specialists in the field that it is also interesting to obtain the same results by using methods of the qualitative theory of ODEs. In the present article, we shall give a simple proof of the result above in this way. An earlier proof of this result of the other authors is essentially more complicated than our one.     

Gradient estimates for ut=ΔF(u) on manifolds and some Liouville-type theorems

In this paper, we first prove a localized Hamilton-type gradient estimate for the positive solutions of Porous Media type equations:$u_t=\mathrm{\Delta }F(u),$ with $F^{\prime }(u)>0$, on a complete Riemannian manifold with Ricci curvature bounded from below. In the second part, we study Fast Diffusion Equation (FDE) and Porous Media Equation (PME):$u_t=\mathrm{\Delta }\left(u^p\right),p>0,$ and obtain localized Hamilton-type gradient estimates for FDE and PME in a larger range of p than that for Aronson–Bénilan estimate, Harnack inequalities and Cauchy problems in the literature. Applying the localized gradient estimates for FDE and PME, we prove some Liouville-type theorems for positive global solutions of FDE and PME on noncompact complete manifolds with nonnegative Ricci curvature, generalizing Yau?s celebrated Liouville theorem for positive harmonic functions.     

H ∞-calculus for the Stokes operator on unbounded domains

We consider the Stokes operator A on unbounded domains of uniform C ^1,1-type. Recently, it has been shown by Farwig, Kozono and Sohr that – A generates an analytic semigroup in the spaces , 1 < q < ∞, where for q ≥ 2 and for q ∈ (1, 2). Moreover, it was shown that A has maximal L ^p -regularity in these spaces for p ∈ (1,∞). In this paper we show that ɛ + A has a bounded H ^∞-calculus in for all q ∈ (1, ∞) and ɛ > 0. This allows to identify domains of fractional powers of the Stokes operator. Received: 12 October 2007     

Decay of solutions for 2D Navier–Stokes equations posed on Lipschitz and smooth bounded and unbounded domains

Initial–boundary value problems for 2D Navier–Stokes equations posed on bounded and unbounded rectangles as well as on bounded and unbounded smooth domains were considered. The existence and uniqueness of regular global solutions in bounded rectangles and bounded smooth domains as well as exponential decay of solutions on bounded and unbounded domains were established.     

Decay of the L∞-norm of solutions of Navier-Stokes equations in unbounded domains

We show that for nn? 4 the L_∞-norm of weak solutions of the Navier-Stokes equations on ?ⁿ with generalized energy inequality decays like $\parallel u(t, \cdot )\parallel _\infty = O(t^{ - ({{n + 1)} \mathord{\left/ {\vphantom {{n + 1)} 2}} \right. \kern-0em} 2}} ),if(1 + | \cdot |)|u(0, \cdot )| \in L_1 $ and $$\int_{\mathbb{R}^n } {u(0,x)} dx = 0$$ . The same holds for strong solutions in all dimensions, if additionally u(0, ·) ε L_p p >n.     

Uniqueness of almost periodic-in-time solutions to Navier�CStokes equations in unbounded domains

We present a uniqueness theorem for almost periodic-in-time solutions to the Navier?CStokes equations in 3-dimensional unbounded domains. Thus far, uniqueness of almost periodic-in-time solutions to the Navier?CStokes equations in unbounded domain, roughly speaking, is known only for a small almost periodic-in-time solution in ${BC(\mathbb {R};L^{3}_w)}$ within the class of solutions that have sufficiently small ${L^{\infty}(L^{3}_w)}$ -norm. In this paper, we show that a small almost periodic-in-time solution in ${BC(\mathbb {R};L^{3}_w\cap L^{6,2})}$ is unique within the class of all almost periodic-in-time solutions in ${BC(\mathbb {R};L^{3}_w\cap L^{6,2})}$ . The proof of the present uniqueness theorem is based on the method of dual equations.     

Pullback attractors for the non-autonomous FitzHugh–Nagumo system on unbounded domains

The existence of a pullback attractor is established for the singularly perturbed FitzHugh–Nagumo system defined on the entire space Rⁿ${\mathbb{R}}^n$ when external terms are unbounded in a phase space. The pullback asymptotic compactness of the system is proved by using uniform a priori estimates for far-field values of solutions. Although the limiting system has no global attractor, we show that the pullback attractors for the perturbed system with bounded external terms are uniformly bounded, and hence do not blow up as a small parameter approaches zero.