José M. Arrieta

Abstract parabolic problems with critical nonlinearities and applications to Navier-Stokes and heat equations

PARABOLIC PROBLEMS WITH CRITICAL NONLINEARITIES AND APPLICATIONS TO NAVIER-STOKES AND HEAT EQUATIONS JOSÉ M. ARRIETA AND ALEXANDRE N. CARVALHO Abstract. We prove a local existence and uniqueness theorem for abstract parabolic problems of the type ẋ = Ax+f(t, x) when the nonlinearity f satisfies certain critical conditions. We apply this abstract result to the Navier-Stokes and heat equations. We prove a local existence and uniqueness theorem for abstract parabolic problems of the type ẋ = Ax+f(t, x) when the nonlinearity f satisfies certain critical conditions. We apply this abstract result to the Navier-Stokes and heat equations.

LINEAR PARABOLIC EQUATIONS IN LOCALLY UNIFORM SPACES

We analyze the linear theory of parabolic equations in uniform spaces. We obtain sharp Lp-Lq-type estimates in uniform spaces for heat and Schrodinger semigroups and analyze the regularizing effect and the exponential type of these semigroups. We also deal with general second-order elliptic operators and study the generation of analytic semigoups in uniform spaces.

Homogenization in a thin domain with an oscillatory boundary

Abstract In this paper we analyze the behavior of the Laplace operator with Neumann boundary conditions in a thin domain of the type R ϵ = { ( x 1 , x 2 ) ∈ R 2 | x 1 ∈ ( 0 , 1 ) , 0 x 2 ϵ G ( x 1 , x 1 / ϵ ) } where the function G ( x , y ) is periodic in y of period L. Observe that the upper boundary of the thin domain presents a highly oscillatory behavior and, moreover, the height of the thin domain, the amplitude and period of the oscillations are all of the same order, given by the small parameter ϵ.

Localization on the Boundary of Blow-up for Reaction-Diffusion Equations with Nonlinear Boundary Conditions

Abstract In this work we analyze the existence of solutions that blow-up in finite time for a reaction–diffusion equation u t − Δu = f(x, u) in a smooth domain Ω with nonlinear boundary conditions ∂u/∂n = g(x, u). We show that, if locally around some point of the boundary, we have f(x, u) = −βu p , β ≥ 0, and g(x, u) = u q then, blow-up in finite time occurs if 2q > p + 1 or if 2q = p + 1 and β < q. Moreover, if we denote by T b the blow-up time, we show that a proper continuation of the blowing up solutions are pinned to the value infinity for some time interval [T, τ] with T b ≤ T < τ. On the other hand, for the case f(x, u) = −βu p , for all x and u, with β > 0 and p > 1, we show that blow-up occurs only on the boundary.

Rates of eigenvalues on a dumbbell domain. Simple eigenvalue case

We obtain the first term in the asymptotic expansion of the eigenvalues of the Laplace operator in a typical dumbbell domain in E2 . This domain consists of two disjoint domains Í2L, iV* joined by a channel Re of height of the order of the parameter e . When an eigenvalue approaches an eigenvalue of the Laplacian in ClL uClR , the order of convergence is £ , while if the eigenvalue approaches an eigenvalue which comes from the channel, the order is weaker: e| lne| . We also obtain estimates on the behavior of the eigenfunctions.

Rapidly varying boundaries in equations with nonlinear boundary conditions. The case of a Lipschitz deformation

We analyze the behavior of solutions of nonlinear elliptic equations with nonlinear boundary conditions of type when the boundary of the domain varies very rapidly. We show that the limit boundary condition is given by , where γ(x) is a factor related to the oscillations of the boundary at point x. For the case where we have a Lipschitz deformation of the boundary, γ is a bounded function and we show the convergence of the solutions in H1 and Cα norms and the convergence of the eigenvalues and eigenfunctions of the linearization around the solutions. If, moreover, a solution of the limit problem is hyperbolic, then we show that the perturbed equation has one and only one solution nearby.

DYNAMICS OF A REACTION-DIFFUSION EQUATION WITH A DISCONTINUOUS NONLINEARITY

We study the nonlinear dynamics of a reaction–diffusion equation where the nonlinearity presents a discontinuity. We prove the upper semicontinuity of solutions and the global attractor with respect to smooth approximations of the nonlinear term. We also give a complete description of the set of fixed points and study their stability. Finally, we analyze the existence of heteroclinic connections between the fixed points, obtaining information on the fine structure of the global attractor.

The Neumann problem in thin domains with very highly oscillatory boundaries

Abstract In this paper, we analyze the behavior of solutions of the Neumann problem posed in a thin domain of the type R ϵ = { ( x 1 , x 2 ) ∈ R 2 ∣ x 1 ∈ ( 0 , 1 ) , − ϵ b ( x 1 ) x 2 ϵ G ( x 1 , x 1 / ϵ α ) } with α > 1 and ϵ > 0 , defined by smooth functions b ( x ) and G ( x , y ) , where the function G is supposed to be l ( x ) -periodic in the second variable y . The condition α > 1 implies that the upper boundary of this thin domain presents a very high oscillatory behavior. Indeed, we have that the order of its oscillations is larger than the order of the amplitude and height of R ϵ given by the small parameter ϵ . We also consider more general and complicated geometries for thin domains which are not given as the graph of certain smooth functions, but rather more comb-like domains.

Elliptic problems in thin domains with highly oscillating boundaries

We study the Laplace operator with Neumann boundary conditions in a 2-dimensional thin domain with a higly oscillating boundary. We obtain the correct limit problem for the case where the boundary is the graph of the oscillating function ϵGϵ(x) where Gϵ(x) = a(x) + b(x)g(x/ϵ) with g periodic and a and b not necessarily constant.

Bifurcation and stability of equilibria with asymptotically linear boundary conditions at infinity

(MS received 18 March 2005; accepted 8 March 2006)We consider an elliptic equation with a nonlinear boundary condition which isasymptotically linear at infinity and which depends on a parameter. As theparameter crosses some critical values, there appear certain resonances in theequation producing solutions that bifurcate from infinity. We study the bifurcationbranches, characterize when they are sub- or supercritical and analyse the stabilitytype of the solutions. Furthermore, we apply these results and techniques to obtainLandesman–Lazer-type conditions guaranteeing the existence of solutions in theresonant case and to obtain an anti-maximum principle.