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In this paper we show that solutions of the heat equation that are given in terms of the heat kernel define semigroups on the family of Frechet spaces L 0 p ( R d ) , the intersection (over all e > 0 ) of the spaces L e p ( R d ) of functions such that ∫ R d e − e | x | 2 | f ( x ) | p d x ∞ . These spaces consist of functions that are ‘large at infinity’, and L 0 1 ( R d ) is the maximal space in which one can use the heat kernel to obtain globally-defined solutions of the heat equation. We prove suitable estimates from L 0 p ( R d ) into L 0 q ( R d ) , q ≥ p , for these semigroups. We then consider the heat semigroup posed in spaces that are dual to these spaces of functions, namely the spaces L − e p ( R d ) of very-rapidly decreasing functions such that ∫ R d e e | x | 2 | f ( x ) | p d x ∞ . We show that ( L p e p ( R d ) ) ′ = L − q e q ( R d ) (with 1 p ∞ and ( p , q ) conjugate), and that the heat flow on L e p ( R d ) is the adjoint of the flow on L − δ q ( R d ) for an appropriate (time-dependent) choice of δ.

In this paper we analyse the asymptotic behaviour of some nonlocal diffusion problems with local reaction term in general metric measure spaces. We find certain classes of nonlinear terms, including logistic type terms, for which solutions are globally defined with initial data in Lebesgue spaces. We prove solutions satisfy maximum and comparison principles and give sign conditions to ensure global asymptotic bounds for large times. We also prove that these problems possess extremal ordered equilibria and solutions, asymptotically, enter in between these equilibria. Finally we give conditions for a unique positive stationary solution that is globally asymptotically stable for nonnegative initial data. A detailed analysis is performed for logistic type nonlinearities. As the model we consider here lack of smoothing effect, important focus is payed along the whole paper on differences in the results with respect to problems with local diffusion, like the Laplacian operator.

In this paper we derive the limiting PDES for parabolic problems where localized large diffusion touches the boundary of the domain. Hence the limit problem reflects the interaction of large diffusion and boundary conditions. The limit problem is a PDE coupled with a system of ODEs through some nonlocal terms. We study the well posedness and dissipativity of the (nonstandard) limit problem as well as the continuity of the dynamics as diffusion becomes large.

In this paper, we analyze evolution problems associated to homogenous operators. We show that they have an homogenous associated semigroup of solutions that must satisfy some sharp estimates when acting on homogenous spaces and on the associated fractional power spaces. These sharp estimates are determined by the homogeneity alone. We also consider fractional diffusion problems and Schrödinger type problems as well. We apply these general results to broad classes of PDE problems including heat or higher order parabolic problems and the associated fractional and Schrödinger problems or Stokes equations. These equations are considered in Lebesgue or Morrey spaces.

In this paper we show how solutions of a wave equation with distributed damping near the boundary converge to solutions of a wave equation with boundary feedback damping. Sufficient conditions are given for the convergence of solutions to occur in the natural energy space.

In this paper we study several PDE problems where certain linear or nonlinear termsin the equation concentrate in the domain, typically (but not exclusively) near the boundary. We analyze some linear and nonlinear elliptic models, linear and nonlinear parabolic ones as well as some damped wave equations. We show that in all these singularly perturbed problems, the concentrating terms give rise in the limit to a modification in the original boundary condition of the problem. Hence we describe in each case which is the singular limit problem and analyze the convergence of solutions.

Linear 2m-th order uniformly elliptic operators are shown to generate semigroups of bounded linear operators with suitable smoothing properties in scales of locally uniform Bessel's and Lebesgue's spaces.

We analyze the asymptotic behavior of positive solutions of parabolic equations with a class of degenerate logistic nonlinearities of the type λ u − n ( x ) u ρ . An important characteristic of this work is that the region where the logistic term n ( ⋅ ) vanishes, that is K 0 = { x : n ( x ) = 0 } , may be non-smooth. We analyze conditions on λ, ρ, n ( ⋅ ) and K 0 guaranteeing that the solution starting at a positive initial condition remains bounded or blows up as time goes to infinity. The asymptotic behavior may not be the same in different parts of K 0 .

Due to the lack of the maximum principle the analysis of higher order parabolic problems in RN is still not as complete as the one of the second-order reaction-diffusion equations. While the critical exponents and then a dissipative mechanism in the subcritical case have already been satisfactorily described (see Cholewa and Rodriguez-Bernal (2012)), for problems in the critical or supercritical regime the questions concerning well or illposedness, as well as possible dissipative properties of the solutions, have not yet been satisfactorily answered. This article is devoted to the analysis of the higher order parabolic problems in R-N in the latter case. Focusing on the critical and supercritical regimes we give sufficient "good"-sign conditions proving that the problem is then globally well posed in L-2(R-N) and even possesses a compact global attractor. On the other hand, for supercritically growing "bad"-signed nonlinearities we show that the problem is ill-posed.

We solve some fourth order parabolic equations, obtained from perturbations of the parabolic bi-Laplacian equation, with special focus on smoothing estimates. Several classes of initial data are considered including data in Lebesgue and Bessel-Lebesgue spaces. Robustness and convergence with respect to the perturbation are also obtained. Also, initial data in large uniform Bessel-Lebesgue spaces are considered as well as equations with higher order powers of the Laplacian.

It is known that the concept of dissipativeness is fundamental for understanding the asymptotic behavior of solutions to evolutionary problems. In this paper we investigate the dissipative mechanism for some semilinear fourth-order parabolic equations in the spaces of Bessel potentials and discuss some weak conditions that lead to the existence of a compact global attractor. While for second-order reaction-diffusion equations the dissipativeness mechanism has already been satisfactorily understood (see Arrieta et al. (2004), doi:10.1142/S0218202504003234), for higher order problems in unbounded domains it has not yet been fully developed. As shown throughout the paper, one of the main differences from the case of reaction-diffusion equations stems from the lack of a maximum principle. Thus we have to rely here on suitable energy estimates for the solutions. As in the case of second-order reaction-diffusion equations, we show here that both linear and nonlinear terms have to collaborate in order to produce dissipativeness. Thus, the dissipative mechanisms in second-order and fourth-order equations are similar, although the lack of a maximum principle makes the proofs more difficult and the results not as complete. Finally, we make essential use of the sharp results of Cholewa and Rodriguez-Bernal (2012), doi:10.1016/j.na.2011.08.022, on linear fourth-order equations with a very large class of linear potentials.

In this paper we consider some fourth order linear and semilinear equations in R N and make a detailed study of the solvability of the Cauchy problem. For the linear equation we consider some weakly integrable potential terms, and for any 1 p ∞ prove that for a suitable family of Bessel potential spaces, H p α ( R N ) , the linear equation defines a strongly continuous analytic semigroup. Using this result, we prove that the nonlinear problems we consider can be solved for initial data in L p ( R N ) and in H p 2 ( R N ) . We also find the corresponding critical exponents, that is, the largest growth allowed for the nonlinear terms for these classes of initial data.

We obtain nonhomogeneous dynamic boundary conditions as a singular limit of a parabolic problem with null flux and potentials and reaction terms concentrating at the boundary.

We analyze the asymptotic behavior of the attractors of a parabolic problem when some reaction and potential terms are concentrated in a neighborhood of a portion Γ of the boundary and this neighborhood shrinks to Γ as a parameter e goes to zero. We prove that the family of attractors is upper continuous at the e=0.

In this paper we study in detail the geometrical structure of global pullback and forwards attractors associated to non-autonomous Lotka-Volterra systems in all the three cases of competition, symbiosis or prey-predator. In particular, under some conditions on the parameters, we prove the existence of a unique non-degenerate global solution for these models, which attracts any other complete bounded trajectory. Thus, we generalize the existence of a unique strictly positive stable (stationary) solution from the autonomous case and we extend to Lotka–Volterra systems the result for scalar logistic equations. To this end we present the sub-supertrajectory tool as a generalization of the now classical sub-supersolution method. In particular, we also conclude pullback and forwards permanence for the above models. Ministerio de Educación y Ciencia Universidad Complutense de Madrid Comunidad Autónoma de Madrid Grupo de Investigación CADEDIF

We consider a reaction diffusion equation ut = Δu + f(x, u) in ℝN with initial data in the locally uniform space [Formula: see text], q ∈ [1, ∞), and with dissipative nonlinearities satisfying s f(x, s) ≤ C(x)s2 + D(x) |s|, where [Formula: see text] and [Formula: see text] for certain [Formula: see text]. We construct a global attractor [Formula: see text] and show that [Formula: see text] is actually contained in an ordered interval [φm, φM], where [Formula: see text] is a pair of stationary solutions, minimal and maximal respectively, that satisfy φm ≤ lim inft→∞ u(t; u0) ≤ lim supt→∞ u(t; u0) ≤ φM uniformly for u0 in bounded subsets of [Formula: see text]. A sufficient condition concerning the existence of minimal positive steady state, asymptotically stable from below, is given. Certain sufficient conditions are also discussed ensuring the solutions to be asymptotically small as |x| → ∞. In this case the solutions are shown to enter, asymptotically, Lebesgue spaces of integrable functions in ℝN, the attractor attracts in the uniform convergence topology in ℝN and is a bounded subset of W2,r(ℝN) for some r > N/2. Uniqueness and asymptotic stability of positive solutions are also discussed. Applications to some model problems, including some from mathematical biology are given.

Lotka-Volterra systems are the canonical ecological models used to analyze population dynamics of competition, symbiosis or prey-predator behaviour involving different interacting species in a fixed habitat. Much of the work on these models has been within the framework of infinite-dimensional dynamical systems, but this has frequently been extended to allow explicit time dependence, generally in a periodic, quasiperiodic or almost periodic fashion. The presence of more general non-autonomous terms in the equations leads to non-trivial difficulties which have stalled the development of the theory in this direction. However, the theory of non-autonomous dynamical systems has received much attention in the last decade, and this has opened new possibilities in the analysis of classical models with general non-autonomous terms. In this paper we use the recent theory of attractors for non-autonomous PDEs to obtain new results on the permanence and the existence of forwards and pullback asymptotically stable global solutions associated to non-autonomous Lotka-Volterra systems describing competition, symbiosis or prey-predator phenomena. We note in particular that our results are valid for prey-predator models, which are not order-preserving: even in the ‘simple’ autonomous case the uniqueness and global attractivity of the positive equilibrium (which follows from the more general results here) is new. Ministerio de Educación y Ciencia Consejería de Innovación, Ciencia y Empresa (Junta de Andalucía) Royal Society University Research Fellowship Grupo de Investigación UCMCAM

We give conditions for the existence of a unique positive complete trajectories for non-autonomous reaction-diffusion equations. Also, attraction properties of the unique complete trajectory is obtained in a pullback sense and also forward in time. As an example, a non-autonomous logistic equation is considered.

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.

We make precise the sense in which spatial homogenization to a constant function in space is attained in a linear parabolic problem when large diffusion in all parts of the domain is assumed. Also interaction between diffusion and boundary flux terms is considered. Our starting point is a detailed analysis of the large diffusion effects on the associated elliptic and eigenvalue problems. Here convergence is shown in the energy space $H^1(\Omega)$ and in the space of continuous functions $C(\overline{\Omega})$. In the parabolic case we prove convergence in the functional space $L^\infty ((0,T),L^2(\Omega)) \bigcap L^2 ((0,T),H^1(\Omega)).

In this paper we show that several known critical exponents for nonlinear parabolic problems are optimal in the sense that supercritical problems are ill posed in a strong sense. We also give an answer to an open problem proposed by Brezis and Cazenave in [9], concerning the behavior of the existence time for critical problems. Our results cover nonlinear heat equations including the case of nonlinear boundary conditions and weigthed spaces settings. In the latter case we show that in some cases the critical exponent is equal to one.

We analyze the linear theory of parabolic equations in uniform spaces. We obtain sharp Lp-Lq-type estimates in uniform spaces for heat and Schrödinger 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.

In this paper we give general and flexible conditions for a reaction diffusion equation to be dissipative in an unbounded domain. The functional setting is based on standard Lebesgue and Sobolev–Lebesgue spaces. We show how the reaction and diffusion mechanisms have to work together to obtain the asymptotic compactness of solutions and therefore the existence of the compact attractor. In particular cases, our results allow us to improve some previous known results.

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 0 and p > 1, we show that blow-up occurs only on the boundary.

We prove the existence of nonconstant stable stationary solutions of an evolution problem with a nonlinear reaction acting on the boundary. These solutions present layers at certain points of the boundary. We also study the behavior of these solutions as the small parameter appearing in the equation approaches zero and show some stability properties of the profiles given by these equilibrium solutions.

In this paper we survey some recent results on the behavior of solutions of parabolic equations subjected to nonlinear boundary conditions. The results range from local existence and regularity of solutions, to global existence, dissipativeness and existence of attractors, and to blow-up in finite time. Some applications are given to some singular perturbation problems and to pattern formation from boundary flux.

The title of the paper says it all. The author considers a reaction-diffusion equation with nonlinear boundary conditions on a bounded domain Ω⊂R N . The initial value u 0 is allowed to be a function in L r (Ω) , with 1

In this paper, we study the asymptotic behavior of solutions for the partly dissipative reaction diffusion equations in n . We prove the asymptotic compactness of the solutions and then establish the existence of the global attractor in 2Ž n. 2Ž n. L L . 2000 Academic Press

The authors study the asymptotic behavior of solutions to a semilinear parabolic problem u t −div(a(x)∇u)+c(x)u=f(x,u) for u=u(x,t), t>0, x∈Ω⊂⊂R N , a(x)>m>0; u(x,0)=u 0 with nonlinear boundary conditions of the form u=0 on Γ 0 , and a(x)∂ n u+b(x)u=g(x,u) on Γ 1 , where Γ i are components of ∂Ω . Under smoothness and growth conditions which ensure the local classical well-posedness of the problem, they indicate some sign conditions under which the solutions are globally defined in time, and somewhat more strong dissipativeness conditions under which they possess a global attractor that captures the asymptotic dynamics of the system. After that the authors study the dependence of the attractors on the diffusion. For a(x)=a e (x) they show their upper semicontinuity on e . Throughout the paper they also pay special attention to the dependence of the estimates obtained on the domain Ω and show that in certain instances the L ∞ bounds on the attractors do not depend on the shape of Ω but rather on |Ω| .

We prove existence, uniqueness and regularity of solutions For heat equations with nonlinear boundary conditions. We study these problems with initial data in L-q(Omega), W-1,W-q(Omega), 1 < q < infinity or measures and with critically growing non-linearities.

The dynamics of a closed thermosyphon are considered. Using an explicit construction, obtained through an inertial manifold, exact low-dimensional models are derived. The behavior of solutions is analyzed for different ranges of the relevant parameters, and the Lorenz model is obtained for a range of parameter values. Numerical experiments are performed for three- and five-mode models.

We will consider the family of wave equations with boundary damping $\qquad\qquad \qquad\qquad \epsilon u_{t t} -\Delta u + \lambda u =f $ on $\Omega \times (0,T)$ $(P_{\epsilon, \lambda, \Gamma_0})\qquad\qquad u_t + \frac{\partial u}{\partial \vec{n}} =g$ on $\Gamma_1 \times (0,T) $ $\qquad\qquad u=0 $ on $\Gamma_0 \times (0,T)$ where $0< \epsilon \leq \epsilon_0$, $\Omega \subset \mathbb R^N$ is a regular open connected set, $\lambda \geq 0$ and $\Gamma = \Gamma_0\cup \Gamma_1$ is a partition of the boundary of $\Omega$. We will also consider the case where $\Gamma_0$ is empty (see below for more precise assumptions on $\lambda$, $\Omega$ and $\Gamma_0$, $\Gamma_1$). For this problem the corresponding formal singular perturbation at $\epsilon =0$ is $\qquad\qquad \qquad\qquad -\Delta u + \lambda u =f$ on $\Omega \times (0,T) $ $(P_{0, \lambda, \Gamma_0}) \qquad\qquad u_t + \frac{\partial u}{\partial \vec{n}} =g$ on $\Gamma_1 \times (0,T) $ $\qquad\qquad u=0 $ on $ \Gamma_0 \times (0,T)$ We are here concerned with the well possedness of both problems for the non--homogeneous case, i.e. $f=f(t,x)$, $g=g(t,x)$, and with the convergence, as $\epsilon$ approaches $0$, of the solutions of $(P_{\epsilon, \lambda, \Gamma_0})$ to solutions of $(P_{0, \lambda, \Gamma_0})$.

We show existence and uniqueness of global solutions for reaction-diffusion equations with almost-monotonic nonlinear terms in L-q(Omega) for each 1

In this paper we study in detail the pullback and forwards attractions to non-autonomous competition Lotka-Volterra system. In particular, under some conditions on the parameters, we prove the existence of a unique non-degenerate global solution for these models, which attracts any other complete bounded trajectory. For that we present the sub-supertrajectory tool as a generalization of the now Classical subsupersolution method. Ministerio de Ciencia y Tecnología Programa de Financiación de Grupos de Investigación UCM-Comunidad de Madrid

In this well-written paper, the authors consider monotone semigroups in ordered spaces and give general results concerning the existence of extremal equilibria and global attractors. \par In the first part of the paper, some notions concerning dissipative systems in ordered space are recalled. Then follow results on the existence of extremal solutions and global attractors and finally on the inclusion of the global attractor in an order interval formed by the minimal and the maximal equilibria. \par In the second part of the paper, they then show some applications of the abstract scheme to various evolutionary problems, from ODEs and retarded functional differential equations to parabolic and hyperbolic PDEs. In particular, they exhibit the dynamical properties of semigroups defined by semilinear parabolic equations in $\Bbb R^N$ with nonlinearities depending on the gradient of the solution. \par The authors consider as well systems of reaction-diffusion equations in $\Bbb R^N$ and provide some results concerning extremal equilibria of the semigroups corresponding to damped wave problems in bounded domains or in $\Bbb R^N$. They further discuss some nonlocal and quasilinear problems, as well as the fourth order Cahn-Hilliard equation.

Summary: "We consider an elliptic equation −Δu+u=0 with nonlinear boundary conditions ∂u/∂n=λu+g(λ,x,u) , where (g(λ,x,s))/s→0 as |s|→∞ . In [Proc. Roy. Soc. Edinburgh Sect. A 137 (2007), no. 2, 225--252; MR2360769 (2009d:35194); J. Differential Equations 246 (2009), no. 5, 2055--2080; MR2494699 (2010c:35016)] the authors proved the existence of unbounded branches of solutions near a Steklov eigenvalue of odd multiplicity and, among other things, provided tools to decide whether the branch is subcritical or supercritical. In this work, we give conditions on the nonlinearity, guaranteeing the existence of a bifurcating branch which is neither subcritical nor supercritical, having an infinite number of turning points and an infinite number of resonant solutions.''

We analyze the asymptotic behavior of the attractors of a parabolic problem when some reaction and potential terms are concentrated in a neighborhood of a portion Γ of the boundary and this neighborhood shrinks to Γ as a parameter e goes to zero. We prove that this family of attractors is upper continuous at e = 0 .

We prove that compact attractors of nonlinear parabolic problems with general potentials have finite fractal and Haussdorf dimension. The linear potentials belong to the space of locally uniform functions in and, unlike other references, they are allowed to change sign.

We show the existence of two special equilibria, the extremal ones, for a wide class of reaction–diffusion equations in bounded domains with several boundary conditions, including non-linear ones. They give bounds for the asymptotic dynamics and so for the attractor. Some results on the existence and/or uniqueness of positive solutions are also obtained. As a consequence, several well-known results on the existence and/or uniqueness of solutions for elliptic equations are revisited in a unified way obtaining, in addition, information on the dynamics of the associated parabolic problem. Finally, we ilustrate the use of the general results by applying them to the case of logistic equations. In fact, we obtain a detailed picture of the positive dynamics depending on the parameters appearing in the equation.

The dynamics of a closed loop thermosyphon are considered. The model assumes a prescribed heat flux along the loop wall and the contribution of axial diffusion. The well-posedness of the model which consists of a coupled ODE and PDE is shown for both the case with diffusion and without diffusion. Boundedness of solutions, the existence of an attractor, and an inertial manifold is proven, and an exact reduction to a low-dimensional model is obtained for the diffusion case. The reduced systems may have far fewer degrees of freedom than the reduction to the inertial manifold. For the three mode models, equivalence with the classical Lorenz equations is shown. Numerical results are presented for five mode models.

We obtain uniform bounds for attractors of parabolic problems with nonlinear boundary conditions with respect to perturbations in the diffusion coefficients. These bounds are the main tool to obtain continuity properties of the attractors relatively to perturbations of the diffusion coefficient. We study two examples where the diffusion is perturbed: one related to homogenization theory and the other taken from composite materials where the diffusion goes to infinity in some subdomain.

In this paper we show that dissipative reaction-diffusion equations in unbounded domains posses extremal semistable ground states equilibria, which bound asymptotically the global dynamics. Uniqueness of such positive ground state and their approximation by extremal equilibria in bounded domains is also studied. The results are then applied to the important case of logistic equations. (C) 2007 Elsevier Inc. All rights reserved.

In this paper we exhibit the dissipative mechanism of the Cahn–Hilliard equation in H1(RN). We show a weak form of dissipativity by showing that each individual solution is attracted, in some sense, by the set of equilibria. We also indicate that strong dissipativity, that is, asymptotic compactness in H1(RN), cannot be in general expected. Then we consider two types of perturbations: a nonlinear perturbation and a small linear perturbation. In both cases we show that, for the resulting equations, the dissipative mechanism becomes strong enough to obtain the existence of a compact global attractor.

The motivations to study the problem considered in this paper come from the theory of composite materials, where the heat diffusion properties can change from one part of the domain to another. Mathematically, this leads to a nonlinear second-order parabolic equation for which the diffusion coefficient becomes large in a subdomain Ω 0 ⊂Ω . The equation is supplemented by a nonlinear boundary condition on ∂Ω and an initial condition. The authors determine the form of the limit problem (the so-called shadow system), which involves an evolution equation for the averages of the density over Ω 0 . The main results include global-in-time existence of solutions and upper semicontinuity of the associated global attractors when the system approaches the shadow system.