Attractors for reaction-diffusion equations on thin domains whose linear part is non-self-adjoint

For a bounded smooth domain Ω⊂R^N_x+N_y let Ω_?, 0<?, be a family of domains squeezed in y∈R^N_y direction. On Ω_? we consider a reaction-diffusion equation with nonsymmetrical linear part. We show that under natural conditions on the nonlinearity the generated semi-flows have global attractors which in a certain sense have limits, as ?↓0.     

Continuity of the dynamics in a localized large diffusion problem with nonlinear boundary conditions

This paper is concerned with singular perturbations in parabolic problems subjected to nonlinear Neumann boundary conditions. We consider the case for which the diffusion coefficient blows up in a subregion Ω₀ which is interior to the physical domain Ω⊂Rn. We prove, under natural assumptions, that the associated attractors behave continuously as the diffusion coefficient blows up locally uniformly in Ω₀ and converges uniformly to a continuous and positive function in .     

Large time behavior of solutions to degenerate parabolic equations with weights

We study the degenerate parabolic equation t_∂u=a(δ(x))upΔu−g(u) in Ω×(0,∞), where Ω⊂RN (N?1) is a smooth bounded domain, p?1, δ(x)=dist(x,∂Ω) and a is a continuous nondecreasing function such that a(0)=0. Under some suitable assumptions on a and g we prove the existence and the uniqueness of a classical solution and we study its asymptotic behavior as t→∞.     

Schauder estimates for oblique derivative problems

We prove that the solution of the oblique derivative parabolic problem in a noncylindrical domain Ω^T belongs to the anisotropic Holder space C^{2+α, 1+α/2}(gwT) 0 < α < 1, even if the nonsmooth “lateral boundary” of Ω^T is only of class C^{1+α, (1+α)/2}). As a corollary, we also obtain an a priori estimate in the Hölder space C^2+α(Ω₀) for a solution of the oblique derivative elliptic problem in a domain Ω₀ whose boundary belongs only to the classe C^1+α.     

On the principal eigenvalue of some nonlocal diffusion problems

In this paper we analyze some properties of the principal eigenvalue λ₁(Ω) of the nonlocal Dirichlet problem (J∗u)(x)−u(x)=−λu(x) in Ω with u(x)=0 in RN?Ω. Here Ω is a smooth bounded domain of RN and the kernel J is assumed to be a C¹ compactly supported, even, nonnegative function with unit integral. Among other properties, we show that λ₁(Ω) is continuous (or even differentiable) with respect to continuous (differentiable) perturbations of the domain Ω. We also provide an explicit formula for the derivative. Finally, we analyze the asymptotic behavior of the decreasing function Λ(γ)=λ₁(γΩ) when the dilatation parameter γ>0 tends to zero or to infinity.     

Dynamics in dumbbell domains III. Continuity of attractors

In this paper we conclude the analysis started in [J.M. Arrieta, A.N. Carvalho, G. Lozada-Cruz, Dynamics in dumbbell domains I. Continuity of the set of equilibria, J. Differential Equations 231 (2006) 551-597] and continued in [J.M. Arrieta, A.N. Carvalho, G. Lozada-Cruz, Dynamics in dumbbell domains II. The limiting problem, J. Differential Equations 247 (1) (2009) 174-202 (this issue)] concerning the behavior of the asymptotic dynamics of a dissipative reaction-diffusion equation in a dumbbell domain as the channel shrinks to a line segment. In [J.M. Arrieta, A.N. Carvalho, G. Lozada-Cruz, Dynamics in dumbbell domains I. Continuity of the set of equilibria, J. Differential Equations 231 (2006) 551-597], we have established an appropriate functional analytic framework to address this problem and we have shown the continuity of the set of equilibria. In [J.M. Arrieta, A.N. Carvalho, G. Lozada-Cruz, Dynamics in dumbbell domains II. The limiting problem, J. Differential Equations 247 (1) (2009) 174-202 (this issue)], we have analyzed the behavior of the limiting problem. In this paper we show that the attractors are upper semicontinuous and, moreover, if all equilibria of the limiting problem are hyperbolic, then they are lower semicontinuous and therefore, continuous. The continuity is obtained in Lp and H¹ norms.     

Existence and non-existence of solutions for a class of Monge-Ampère equations

We study the boundary value problems for Monge-Ampère equations: detD²u=e−u in Ω⊂Rn, n?1, u_|∂Ω=0. First we prove that any solution on the ball is radially symmetric by the argument of moving plane. Then we show there exists a critical radius such that if the radius of a ball is smaller than this critical value there exists a solution, and vice versa. Using the comparison between domains we can prove that this phenomenon occurs for every domain. Finally we consider an equivalent problem with a parameter detD²u=e−tu in Ω, u_|∂Ω=0, t?0. By using Lyapunov-Schmidt reduction method we get the local structure of the solutions near a degenerate point; by Leray-Schauder degree theory, a priori estimate and bifurcation theory we get the global structure.     

Some blow-up problems for a semilinear parabolic equation with a potential

The blow-up rate estimate for the solution to a semilinear parabolic equation ut=Δu+V(x)|u|^p−1u in Ω×(0,T) with 0-Dirichlet boundary condition is obtained. As an application, it is shown that the asymptotic behavior of blow-up time and blow-up set of the problem with nonnegative initial data u(x,0)=Mφ(x) as M goes to infinity, which have been found in [C. Cortazar, M. Elgueta, J.D. Rossi, The blow-up problem for a semilinear parabolic equation with a potential, preprint, arXiv: math.AP/0607055, July 2006], is improved under some reasonable and weaker conditions compared with [C. Cortazar, M. Elgueta, J.D. Rossi, The blow-up problem for a semilinear parabolic equation with a potential, preprint, arXiv: math.AP/0607055, July 2006].     

Boundary behaviour of the unique solution to a singular Dirichlet problem with a convection term

By Karamata regular variation theory and constructing comparison functions, we derive that the boundary behaviour of the unique solution to a singular Dirichlet problem −Δu=b(x)g(u)+λq|∇u|, u>0, x∈Ω, u_|∂Ω=0, which is independent of λq|∇uλ|, where Ω is a bounded domain with smooth boundary in RN, λ∈R, q∈(0,2], lims→⁰⁺g(s)=+∞, and b is non-negative on Ω, which may be vanishing on the boundary.     

Two-sided estimates of a solution of the Neumann problem as t ← ∠ for a second-order Quasilinear Parabolic Equation

We establish exact upper and lower bounds as t ← ∠ for the norm ‖u(·, t)‖ _{L ∞(Ω)} of a solution of the Neumann problem for a second-order quasilinear parabolic equation in the region D=Ω×{>0}, where Ω is a region with noncompact boundary.     

Existence and nonexistence of blow-up boundary solutions for sublinear elliptic equations

In this paper we consider the semilinear elliptic problem Δu=a(x)f(u), u?0 in Ω, with the boundary blow-up condition u_|∂Ω=+∞, where Ω is a bounded domain in RN(N?2), a(x)∈C(Ω) may blow up on ∂Ω and f is assumed to satisfy (f₁) and (f₂) below which include the sublinear case f(u)=um, m∈(0,1). For the radial case that Ω=B (the unit ball) and a(x) is radial, we show that a solution exists if and only if . For Ω a general domain, we obtain an optimal nonexistence result. The existence for nonradial solutions is also studied by using sub-supersolution method.     

On nonlinear diffusion problems with strong degeneracy

In this paper, we study the “triply” degenerate problem: bt(v)−Δg(v)+divΦ(v)=f on Q:=(0,T)×Ω, b(v(0,⋅))=b(v₀) on Ω and “g(v)=g(a) on some part of the boundary (0,T)×∂Ω,” in the case of continuous nonhomogeneous and nonstationary boundary data a. The functions b,g are assumed to be continuous, locally Lipschitz, nondecreasing and to verify the normalization condition b(0)=g(0)=0 and the range condition R(b+g)=R. Using monotonicity and penalization methods, we prove existence of a weak renormalized entropy solution in the spirit of [K. Ammar, J. Carrillo, P. Wittbold, Scalar conservation laws with general boundary condition and continuous flux function, J. Differential Equations 228 (2006) 111-139].     

Reconstruction of two time independent coefficients in an inverse problem for a phase field system

In this paper we present stability results concerning the inverse problem of determining two time independent coefficients for a phase field system in a bounded domain Ω⊂Rⁿ for the dimension n≤3 with a single observation on a subdomain ω?Ω and the Sobolev norm of certain partial derivatives of the solutions at a fixed positive time θ∈(0,T) over the whole spatial domain. The proof of these results relies on an appropriate Carleman estimate for the phase field system.     

Optimal Control of the Obstacle for a Parabolic Variational Inequality

An optimal control problem for a parabolic obstacle variational inequality is considered. The obstacle in L²(0, T; H²(Ω) ∩ H¹₀(Ω)) with ψ_t ∈ L²(Q) is taken as the control, and the solution to the obstacle problem is taken as the state. The goal is to find the optimal control so that the state is close to the desired profile while the norm of the obstacle is not too large. Existence and necessary conditions for the optimal control are established.     

Formulas of Liapunov functions for systems of linear partial differential equations

Formulas of explicit quadratic Liapunov functions for showing asymptotic stability of the system of linear partial differential equations on (0,∞)×Ω, are constructed, where A is an n×n real matrix, u=T(u₁,u₂,…,un), Ω is a bounded domain in Rk with smooth boundary ∂Ω, and Δ denotes the Laplacian operator on Rk with Δu=T(Δu₁,Δu₂,…,Δun). These formulas are also modified and applied to a number of nonautonomous linear and nonlinear systems and models in structural stability, traveling wave, and Navier-Stokes equations.     

Halbgruppen und semilineare Anfangs-Randwertprobleme

A semilinear parabolic initial-boundary-value problem of order 2m in a possibly unbounded domain Ωx(O,T), Ω?Rⁿ, is considered within the framework of the L_p-and C^α-theory. In the first case a proof is given of the existence of a “strict” solution of the corresponding evolution equation. In the second case one can guarantee a classical solution, provided the homogeneous linear parabolic equation has a unique classical solution. Only local solvability is considered. The nonlinearity is a Hölder-continuous function of the derivatives up to the order 2m-1 of the unknown solution. The principal tool is the semigroup-theory in L_p(Ω) as well as in C^α( \(\bar \Omega \) ). In the latter case the semigroup is not strongly continuous, but it has sufficiently good properties to use it for existence proofs of classical solutions.     

A sufficiency class for global (in time) solutions to the 3D Navier-Stokes equations

Let Ω be an open domain of class C² contained in R³, let L²(Ω)³ be the Hilbert space of square integrable functions on Ω and let H[Ω]?H be the completion of the set, , with respect to the inner product of L²(Ω)³. A well-known unsolved problem is that of the construction of a sufficient class of functions in H which will allow global, in time, strong solutions to the three-dimensional Navier-Stokes equations. These equations describe the time evolution of the fluid velocity and pressure of an incompressible viscous homogeneous Newtonian fluid in terms of a given initial velocity and given external body forces. In this paper, we use the analytic nature of the Stokes semigroup to construct an equivalent norm for H, which provides strong bounds on the nonlinear term. This allows us to prove that, under appropriate conditions, there exists a number u₊, depending only on the domain, the viscosity, the body forces and the eigenvalues of the Stokes operator, such that, for all functions in a dense set D contained in the closed ball B(Ω)?B of radius in H, the Navier-Stokes equations have unique, strong, solutions in C¹((0,∞),H).     

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.     

Stationary isothermic surfaces and some characterizations of the hyperplane in the N-dimensional Euclidean space

We consider the entire graph S of a continuous real function over RN−1 with N?3. Let Ω be a domain in RN with S as a boundary. Consider in Ω the heat flow with initial temperature 0 and boundary temperature 1. The problem we consider is to characterize S in such a way that there exists a stationary isothermic surface in Ω. We show that S must be a hyperplane under some general conditions on S. This is related to Liouville or Bernstein-type theorems for some elliptic Monge-Ampère-type equation.     

On an eigenvalue problem involving variable exponent growth conditions

We study the problem in Ω, u=0 on ∂Ω, where Ω is a bounded domain in R^N, is a continuous function and λ and ε are two positive constants. We prove that for any ε>0 each λ∈(0,λ₁) is an eigenvalue of the above problem, where λ₁ is the principal eigenvalue of the Laplace operator on Ω. Moreover, for each eigenvalue λ∈(0,λ₁) it corresponds a unique eigenfunction. The proofs will be based on the Banach fixed point theorem combined with adequate variational techniques.