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Abstract This paper deals with the Neumann problem for a fully parabolic chemotaxis–haptotaxis model of cancer invasion given by { u t = Δ u − χ ∇ ⋅ (u ∇ v) − ξ ∇ ⋅ (u ∇ w) + u (a − μ u r − 1 − λ w) , x ∈ Ω , t > 0 , τ v t = Δ v − v + u , x ∈ Ω , t > 0 , w t = − v w , x ∈ Ω , t > 0. Here, Ω ⊂ R N (N ≥ 1) is a bounded domain with smooth boundary and τ > 0 , r > 1 , λ ≥ 0 , a ∈ R , μ , ξ and χ are positive constants. It is shown that the corresponding initial–boundary value problem possesses a unique global bounded classical solution in the cases r > 2 or r = 2 , with μ > μ ⁎ = (N − 2) + N (χ + C β) C N 2 + 1 1 N 2 + 1 for some positive constants C β and C N 2 + 1. Furthermore, the large time behavior of solutions to the problem is also investigated. Specially speaking, when a is appropriately large, the corresponding solution of the system exponentially decays to ((a μ) 1 r − 1 , (a μ) 1 r − 1 , 0) if μ is large enough. This result improves or extends previous results of several authors. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we consider an initial-boundary value problem of the p-Laplacian parabolic equation { u t (x , t) = div (| ∇ u (x , t) | p − 2 ∇ u (x , t)) + f (u (x , t)) , (x , t) ∈ Ω × (0 , + ∞) , u (x , t) = 0 , (x , t) ∈ ∂ Ω × [ 0 , + ∞) , u (x , 0) = u 0 ≥ 0 , x ∈ Ω ‾ , where p ≥ 2 and Ω is a bounded domain of R N (N ≥ 1) with smooth boundary ∂Ω. The main contribution of this work is to introduce a new condition (C p) α ∫ 0 u f (s) d s ≤ u f (u) + β u p + γ , u > 0 for some α , β , γ > 0 with 0 < β ≤ (α − p) λ 1 , p p , where λ 1 , p is the first eigenvalue of p-Laplacian Δ p , and we use the concavity method to obtain the blow-up solutions to the above equations. In fact, it will be seen that the condition (C p) improves the conditions ever known so far. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we are interested in the global existence and large-time behavior of solutions to the initial-boundary value problem for subcritical Kuramoto–Sivashinsky-type equationwhere the nonlinear term depends on the unknown function u and its derivative and satisfy the estimatewith such that The aim of this paper is to prove the global existence of solutions to the initial-boundary value problem (0.1) in subcritical case, when the nonlinear term has a time decay rate less than that of the linear terms of Eq. (0.1). Also we find the main term of the asymptotic representation of solutions. [Copyright &y& Elsevier]

Abstract: The paper deals with canard solutions at very general turning points of smooth singular perturbation problems in two dimensions. We follow a geometric approach based on the use of -normal forms, centre manifolds and (family) blow up, as we did in (Trans. Amer. Math. Soc., to appear). In (Trans. Amer. Math. Soc., to appear) we considered the existence of manifolds of canard solutions for given appropriate boundary conditions. These manifolds need not be smooth at the turning point. In this paper we essentially study the transition time along such manifolds, as well as the divergence integral, providing a structure theorem for these integrals. As a consequence we get a nice structure theorem for the transition equation, governing the canard solutions. It permits to compare different control manifolds and to obtain a precise description of the entry–exit relation of different canard solutions. Attention is also given to the special case in which the canard manifolds are smooth, i.e. when “formal” canard solutions exist. [Copyright &y& Elsevier]

Abstract: In this work a general class of nonlinear abstract equations satisfying a generalized strong maximum principle is considered in order to study the behavior of the bounded components of positive solutions bifurcating from the curve of trivial states at a nonlinear eigenvalue with geometric multiplicity one. Since the unilateral theorems of Rabinowitz (J. Funct. Anal. 7 (1971) 487, Theorems 1.27 and 1.40) are not true as originally stated (cf. the very recent counterexample of Dancer, Bull. London Math. Soc. 34 (2002) 533), in order to get our main results the unilateral theorem of López-Gómez (Spectral Theory and Nonlinear Functional Analysis, Research Notes in Mathematics, vol. 426, CRC Press, Boca Raton, FL, 2001, Theorem 6.4.3) is required. Our analysis fills some serious gaps existing is some published papers that were provoked by a direct use of Rabinowitz''s unilateral theory. Actually, the abstract theory developed in this paper cannot be covered with the pioneering results of Rabinowitz (1971), since in Rabinowitz''s context any component of positive solutions must be unbounded, by a celebrated result attributable to Dancer (Arch. Rational Mech. Anal. 52 (1973) 181). [Copyright &y& Elsevier]

Abstract: We address the analysis of a nonlinear and degenerating PDE system, proposed by M. Frémond for modelling phase transitions in viscoelastic materials subject to thermal effects. The system features an internal energy balance equation, governing the evolution of the absolute temperature ϑ, an evolution equation for the phase change parameter χ, and a stress–strain relation for the displacement variable u. The main novelty of the model is that the equations for χ and u are coupled in such a way as to take into account the fact that the properties of the viscous and of the elastic parts influence the phase transition phenomenon in different ways. However, this brings about an elliptic degeneracy in the equation for u which needs to be carefully handled. In this paper, we first prove a local (in time) well-posedness result for (a suitable initial–boundary value problem for) the above mentioned PDE system, in the (spatially) three-dimensional setting. Secondly, we restrict to the one-dimensional case, in which, for the same initial–boundary value problem, we indeed obtain a global well-posedness theorem. [Copyright &y& Elsevier]

Abstract: In this paper, we establish the global attractivity of the positive steady state of the diffusive Nicholson''s equation with homogeneous Neumann boundary value under a condition that makes the equation a non-monotone dynamical system. To achieve this, we develop a novel method: combining a dynamical systems argument with maximum principle and some subtle inequalities. [Copyright &y& Elsevier]

Abstract: Let Ω be a bounded domain in , , with the boundary . We consider the following singularly perturbed nonlinear elliptic problem on Ω where ν is an exterior normal to ∂Ω and a nonlinearity f of subcritical growth. Under rather strong conditions on f, it has been known that for small , there exists a solution of the above problem which exhibits a spike layer near local maximum points of the mean curvature H on ∂Ω as . In this paper, we obtain the same result under some conditions on f (Berestycki–Lions conditions), which we believe to be almost optimal. [Copyright &y& Elsevier]

Abstract: In this paper we consider the one-dimensional p-Laplacian boundary value problem on time scales where is p-Laplacian operator, i.e., , . Some new results are obtained for the existence of at least single, twin or triple positive solutions of the above problem by using Krasnosel''skii''s fixed point theorem, new fixed point theorem due to Avery and Henderson and Leggett–Williams fixed point theorem. This is probably the first time the existence of positive solutions of one-dimensional p-Laplacian boundary value problems on time scales has been studied. [Copyright &y& Elsevier]

Abstract: In this paper we give a new perspective on the Cauchy integral and transform and Hardy spaces for Dirac-type operators on manifolds with corners of codimension two. Instead of considering Banach or Hilbert spaces, we use polyhomogeneous functions on a geometrically “blown-up” version of the manifold called the total boundary blow-up introduced by Mazzeo and Melrose [R.R. Mazzeo, R.B. Melrose, Analytic surgery and the eta invariant, Geom. Funct. Anal. 5 (1) (1995) 14–75]. These polyhomogeneous functions are smooth everywhere on the original manifold except at the corners where they have a “Taylor series” (with possible log terms) in polar coordinates. The main application of our analysis is a complete Fredholm theory for boundary value problems of Dirac operators on manifolds with corners of codimension two. [Copyright &y& Elsevier]

Abstract: The paper concerns the well-posedness problem of an evolutionary weighted p-Laplacian with boundary degeneracy. Different from the classical theory for linear equations, it is shown that the degenerate portion of the boundary should be decomposed into two parts: the strongly degenerate boundary on which the equation exhibits hyperbolic characteristics and the weakly degenerate boundary on which the equation still exhibits parabolic characteristics. We formulate reasonably the boundary value condition and establish the existence and uniqueness theorems. [Copyright &y& Elsevier]

Abstract: We analyze the dynamics of a reaction–diffusion equation with homogeneous Neumann boundary conditions in a dumbbell domain. We provide an appropriate functional setting to treat this problem and, as a first step, we show in this paper the continuity of the set of equilibria and of its linear unstable manifolds. [Copyright &y& Elsevier]

Abstract: In this paper, some sufficient conditions for the existence of solutions to the boundary value problems of a class of second order difference equation are obtained by using the critical point theory. [Copyright &y& Elsevier]

Abstract: In this paper, we present global existence results for the following problem where , a positive parameter and h a nonnegative measurable function on which may be singular at and/or , and with . By applying the global bifurcation theorem and figuring the shape of unbounded subcontinua of solutions, we obtain many different types of global existence results of positive solutions. We also obtain existence results of sign-changing solutions for when f is an odd symmetric function. [Copyright &y& Elsevier]

Abstract: In this paper we study bifurcation from simple eigenvalues for systems of differential equations; we prove the existence of global bifurcating branches of solutions on which the Maslov index of suitable associated linear systems is preserved. [Copyright &y& Elsevier]

Abstract: We show in this paper that there exist domains Ω, which are not conformally equivalent to starshaped domains, such that the Dirichlet problem: , in Ω; on ∂Ω has no solution. Some nonexistence results about the Dirichlet problem: , in Ω; on ∂Ω for certain over three dimensional non-starshaped domains are also obtained. [Copyright &y& Elsevier]

Abstract: In this paper, we consider the initial-boundary value problem of a semilinear parabolic equation with local and non-local (localized) reactions in a ball: in where and . If , there exist blow-up solutions of this problem for large initial data. We treat the radially symmetric and one peak non-negative solution of this problem. We give the complete classification of total blow-up phenomena and single point blow-up phenomena according to p and q. [(i)] If or , then single point blow-up occurs whenever solutions blow up. [(ii)] If , both phenomena, total blow-up and single point blow-up, occur depending on the initial data. [(iii)] If , total blow-up occurs whenever solutions blow up. [(iv)] If , every solution exists globally in time. [Copyright &y& Elsevier]

Abstract: In this paper, one-dimensional () nonlinear Schrödinger equationwith Periodic Boundary Conditions is considered; is a real parameter and the nonlinearityis a real analytic function in a neighborhood of the origin. The KAM machinery is adapted to fit the above equation so as to construct small-amplitude periodic or quasi-periodic solutions corresponding to finite dimensional invariant tori for an associated infinite dimensional dynamical system. [Copyright &y& Elsevier]