Browsing by Browse by FOR 2008 "010110 Partial Differential Equations"

A spatially heterogeneous reaction-diffusion system modelling predator-prey interaction is studied, where the interaction is governed by a Holling type II functional response. Existence of multiple positive steady states and global bifurcation branches are examined as well as related dynamical behavior. It is found that while the predator population is not far from a constant level, the prey population could be extinguished, persist or blow up depending on the initial population distributions, the various parameters in the system, and the heterogeneous environment. In particular, our results show that when the prey growth is strong, the spatial heterogeneity of the environment can play a dominant role for the presence of the Allee effect. Our mathematical analysis relies on bifurcation theory, topological methods, various comparison principles and elliptic estimates. We combine these methods with monotonicity arguments to the system through the use of some new auxiliary scalar equations, though the system itself does not keep an order structure as the competition system does. Among other things, this allows us to obtain partial descriptions of the dynamical behavior of the system.

We consider the following problem, [**EQUATION**] where μ > 0 is a large parameter, Ω is a bounded domain in ℝⁿ, N ≤ 3 and 2* = 2N/(N - 2). Let H(P) be the mean curvature function of the boundary. Assuming that H(P) has a local minimum point with positive minimum, then for any integer k, the above problem has a k-boundary peaks solution. As a consequence, we show that if Ω is 'strictly convex', then the above problem has arbitrarily many solutions, provided that μ is large.

We study a nonlinear diffusion equation of the form ut = uxx + f (u) (x ε [g(t), h(t)]) with free boundary conditions g'(t) = -ux(t, g(t)) + α and h'(t) = -ux(t, h(t)) - α for some α > 0. Such problems may be used to describe the spreading of a biological or chemical species, with the free boundaries representing the expanding fronts. When α = 0, the problem was recently investigated by Du and Lin (SIAM J Math Anal 42:377-405, 2010) and Du and Lou (J Euro Math Soc arXiv:1301.5373). In this paper we consider the case α > 0. In this case shrinking (i.e. h(t)-g(t) → 0) may happen, which is quite different from the case α = 0. Moreover, we show that, under certain conditions on f, shrinking is equivalent to vanishing (i.e. u → 0), both of them happen as t tends to some finite time. On the other hand, every bounded and positive time-global solution converges to a nonzero stationary solution as t → ∞. As applications, we consider monostable, bistable and combustion types of nonlinearities, and obtain a complete description on the asymptotic behavior of the solutions.

Let B₁(0) 'C RN' be the unit ball centred at the origin, N ≥ 3. In this paper, we analyse the profile of the ground state solution of the Hénon equation - ∆u = │x│'au⁻¹ in B₁ (0), u = 0 on ∂B₁ (0). We prove that for fixed p ε (2,2*), (2* = 2N/(N - 2)), the maximum point xₐ of the ground state solution uₐ satisfies a(1 - │xₐ│) → l ε(0,+ ∞) as a → ∞. We also obtain the asymptotic behaviour of uₐ, which shows that the ground state solution is non-radial. Moreover, we prove the existence of multi-peaked solutions and give their asymptotic behaviour.

We give a complete description of the long-time asymptotic profile of the solution to a free boundary model considered recently in [10]. This model describes the spreading of an invasive species in an environment which shifts with a constant speed, and the research of [10] indicates that the species may vanish, or spread successfully, or fall in a borderline case. In the case of successful spreading, the long-time behavior of the population is not completely understood in [10]. Here we show that the spreading of the species is governed by two traveling waves, one has the speed of the shifting environment, giving the profile of the retreating tail of the population, while the other has a faster speed determined by a semi-wave, representing the profile of the advancing front of the population.

This paper is concerned with a diffusive Lotka-Volterra type competition system with a free boundary in one space dimension. Such a system may be used to describe the invasion of a new species into the habitat of a native competitor, and its long-time dynamical behavior can be described by a spreading-vanishing dichotomy. The main purpose of this paper is to determine the asymptotic spreading speed of the invading species when its spreading is successful, which involves two systems of traveling wave type equations.

We consider two cases of the asymptotically flat scalar-flat Yamabe problem on a non-compact manifold with inner boundary in dimension n ≥ 3. First, following arguments of Cantor and Brill in the compact case, we show that given an asymptotically flat metric g, there is a conformally equivalent asymptotically flat scalar-flat metric that agrees with g on the boundary. We then replace the metric boundary condition with a condition on the mean curvature: given a function f on the boundary that is not too large, we show that there is an asymptotically flat scalar-flat metric, conformally equivalent to g whose boundary mean curvature is given by f. The latter case involves solving an elliptic PDE with critical exponent using the method of sub- and supersolutions. Both results require the usual assumption that the Sobolev quotient is positive.

In this paper, we study the boundary behavior of positive solutions of the following equation...

We consider the following nonlinear Schrödinger equation in R3...

The Chern-Simons theories were developed to explain certain condensed matter phenomena, anyon physics, superconductivity, quantum mechanics and so on. In this paper, we will study the (2 + 1)-dimensional relativistic Abelian Chern-Simons-Higgs model on a torus Ω.

In the last few decades, various Chern-Simons field theories have been studied, largely motivated by their applications to the physics of high-critical-temperature superconductivity. These Chern-Simons theories can be reduced to systems of nonlinear partial differential equations, which have posed many mathematically challenging problems for analysts. For the abelian case, the relativistic Chern-Simons model was proposed by Jakiw and Weinberg [10] and by Hong, Kim, and Pac [9]. The energy minimizer of this model satisfies a Bogomol'nyĭ-type system of first-order differential equations.

In this paper, we report some recent theoretical work of Du-Shi, Du-Liang and Du-Peng-Wang that examines the effect of environmental changes in three well-known diffusive population models, each modeling the interaction of two population species in a common habitat. For each model, we consider the situation that one of the two species is endangered and a simple protection zone is created in the habitat for the endangered species. The focus is on the effect of the protection zone on the dynamics of the two species systems. We demonstrate that, though the effects of the protection zone on the dynamical behavior of the three systems are significantly different from each other, they all share one important property, namely, there exists a critical patch size for the protection zone: If the protection zone is below this size, each model behaves similarly to the no-protection zone case, but every model undergoes profound changes in dynamical behavior once the protection zone is above the critical patch size, and in such a case the endangered species is always saved from extinction.

We study the Cauchy problem ut = uxx + f(u) (t > 0, x ∈ ℝ¹), u(0, x) = uₒ(x) (x ∈ ℝ¹), where f(u) is a locally Lipschitz continuous function satisfying f(0) = 0. We show that any nonnegative bounded solution with compactly supported initial data converges to a stationary solution as t → ∞. Moreover, the limit is either a constant or a symmetrically decreasing stationary solution. We also consider the special case where f is a bistable nonlinearity and the case where f is a combustion type nonlinearity. Examining the behavior of a parameter-dependent solution uλ, we show the existence of a sharp threshold between extinction (i.e., convergence to 0) and propagation (i.e., convergence to 1). The result holds even if f has a jumping discontinuity at u = 1.

This paper is concerned with the solitary wave solutions for a generalized quasilinear Schrödinger equation in RN involving critical exponents, which have appeared from plasma physics, as well as high-power ultashort laser in matter. We find the related critical exponents for a generalized quasilinear Schrödinger equation and obtain its solitary wave solutions by using a change of variables and variational argument.

We study O(d)-equivariant biharmonic maps in the critical dimension. A major consequence of our study concerns the corresponding heat flow. More precisely, we prove that blowup occurs in the biharmonic map heat flowfrom B⁴(0, 1) into S⁴. To our knowledge, this was the first example of blowup for the biharmonic map heat flow. Such results have been hard to prove, due to the inapplicability of the maximum principle in the biharmonic case. Furthermore, we classify the possible O(4)-equivariant biharmonic maps from R⁴ into S⁴, and we show that there exists, in contrast to the harmonic map analogue, equivariant biharmonic maps from B⁴(0, 1) into S⁴ that wind around S⁴ as many times as we wish. We believe that the ideas developed herein could be useful in the study of other higher-order parabolic equations.

This paper describes differential convolution and its application to high-resolution three dimensional medical images to distinguish between normal and diseased tissue. Underlying image processing principles are presented focusing on texture analysis and prototype application of three dimensional convolution to identify normal, benign and malignant tissue.

In this paper we consider the diffusive competition model consisting of an invasive species with density 'u' and a native species with density 'v', in a radially symmetric setting with free boundary.

We study the diffusive logistic equation with a free boundary in time-periodic environment. Such a model may be used to describe the spreading of a new or invasive species, with the free boundary representing the expanding front. For time independent environment, in the cases of one space dimension, and higher space dimensions with radial symmetry, this free boundary problem has been studied in Du and Lin (2010), Du and Guo (2011). In both cases, a spreading-vanishing dichotomy was established, and when spreading occurs, the asymptotic spreading speed was determined. In this paper, we show that the spreading-vanishing dichotomy is retained in time-periodic environment, and we also determine the spreading speed. The former is achieved by further developing the earlier techniques, and the latter is proved by introducing new ideas and methods.

We introduce and study a class of free boundary models with "nonlocal diffusion", which are natural extensions of the free boundary models in [16]and elsewhere, where “local diffusion” is used to describe the population dispersal, with the free boundary representing the spreading front of the species. We show that this nonlocal problem has a unique solution defined for all time, and then examine its long-time dynamical behavior when the growth function is of Fisher-KPP type. We prove that a spreading-vanishing dichotomy holds, though for the spreading-vanishing criteria significant differences arise from the well known local diffusion model in [16].

In this paper, we consider the diffusive Leslie predator-prey model with large intrinsic predator growth rate, and investigate the change of behavior of the model when a simple protection zone Ω₀ for the prey is introduced. As in earlier work [Y. Du, J. Shi. A diffusive predator-prey model with a protection zone. J. Differential Equations 229 [2006] 63-91: Y. Du. X. Liang. A diffusive competition model with a protection zone. J. Differential Equations 244 (2008) 61-86] we show the existence of a critical patch size of the protection zone, determined by the first Dirichlet eigenvalue of the Laplacian over Ω₀ and the intrinsic growth rate of the prey, so that there is fundamental change of the dynamical behavior of the model only when Ω₀ is above the critical patch size. However, our research here reveals significant difference of the model's behavior from the predator-prey model studies in [Y. Du, J. Shi, A diffusive predator-prey model with a protection some, J. Differential Equations 229 (2006) 63-91] with the same kind of protection zone. We show that the asymptotic profile of the population distribution of the Leslie model is governed by a standard boundary blow-up problem, and classical or degenerate logistic equations.

We study the competition model where the coefficient functions are strictly positive over the underlying spatial region Ω except b(x), which vanishes in a nontrivial subdomain of Ω, and is positive in the rest of Ω. We show that there exists a critical number λ* such that if λ <λ*, then the model behaves similarly to the well-studied classical competition model where all the coefficient functions are positive constants, but when λ>λ*, new phenomena occur. Our results demonstrate the fact that heterogeneous environmental effects on population models are not only quantitative, but can be qualitative as well. In part I here, we mainly study two kinds of steady-state solutions which determine the dynamics of the model: one consists of finite functions while the other consists of generalized functions which satisfy (u, v)=(∞, 0) on the part of the domain that b(x) vanishes, but are positive and finite on the rest of the domain, and are determined by certain boundary blow-up systems. The research is continued in part II, where these two kinds of steady-state solutions will be used to determine the dynamics of the model.

This is the second part of our study on the competition model where the coefficient functions are strictly positive over the underlying spatial region Ω except b(x), which vanishes in a nontrivial subdomain of Ω, and is positive in the rest of Ω. In part I, we mainly discussed the existence of two kinds of steady-state solutions of this system, namely, the classical steady-states and the generalized steady-states. Here we use these solutions to determine the dynamics of the model. We do this with the help of the perturbed model where b(x) is replaced by b(x)+ε, which itself is a classical competition model. This approach also reveals the interesting relationship between the steady-state solutions (both classical and generalized) of the above system and that of the perturbed system.

To demonstrate the influence of spatial heterogeneity on the predator-prey model, we study the effects of the partial vanishing of the non negative coefficient functions b(x) and e(x), respectively, in the steady-state predator-prey model -d₁(x)∆u=λa₁(x)u-b(x)u²-c(x)UV, d₂(x)∆v-µa²(x)c-e(x) vˆ²+d(x)UV, u|∂Ω=v|_∂Ω=0, where all other coefficient functions are strictly positive over the bounded domain Ω in RN. Critical values of the parameter λ are obtained to show that, in each case, the vanishing has little effect on the behavior of the model when λ is below the critical value, while essential changes occur once λ is beyond the critical value.

In this paper, we consider the following problem involving fractional Laplacian operator:(−Δ)αu=|u|2∗α-2-εu+λu in Ω, u = 0 on ∂Ω,(1) where Ω is a smooth bounded domain in RN, ε ∈[0, 2∗α−2), 0 < α < 1, 2∗α = 2N N-2α, and (−Δ)α is either the spectral fractional Laplacian or the restricted fractional Laplacian. We show for problem (1) with the spectral fractional Laplacian that for any sequence of solutions un of (1) corresponding to εn ∈ [0, 2∗α - 2), satisfying ǁunǁH ≤ C in the Sobolev space H defined in (1.2), un converges strongly in H provided that N > 6α and λ > 0. The same argument can also be used to obtain the same result for the restricted fractional Laplacian. An application of this compactness result is that problem (1) possesses infinitely many solutions under the same assumptions.

Introduction: Cancer care workers experience high levels of occupational stress that can have adverse mental and physical health consequences. Educating health professionals about self-care practices throughout their careers can potentially build resilience. Our study aimed to evaluate the effects of an educational intervention to improve recovery from job stress, increase satisfaction with current self-care practices and improve sleep quality. Methods: An equivalent, randomised comparison, pretest-post-test intervention design was used to investigate the effects of a 1-day workshop (plus educational material) compared with written educational material alone, on measures of recovery experiences (i.e. psychological detachment from work, relaxation, mastery experiences and control over leisure), satisfaction with recovery-related self-care practices and perceived sleep quality of 70 cancer care workers. Results: Workshop participants reported greater mean changes 6 weeks postworkshop for total recovery experiences (F(1,69) = 8.145, P = .008), selfcare satisfaction (F(1,69) = 8.277, P = .005) and perceived sleep quality (F(1,69) = 9.611, P = .003). There was a decline in the scores of the control group over the 6-week period for all measures. Workshop participants not only avoided this decline, but demonstrated increased mean scores, with a significant main effect 6 weeks post-workshop, compared with the control group (F(3,63) = 4.262, P = .008). Conclusions: A 1-day intervention workshop improved recovery skills, satisfaction with self-care practices and perceived sleep quality of oncology nurses and radiation therapists. Outcomes were enhanced when participants actively participated in experiential group-based learning compared with receiving written material alone. This intervention has the potential to enhance resilience and prevent burnout at different points in a cancer worker's career.

In this paper, we study the singular behavior at x=0of positive solutions to the equation −u = λ |x|2 u − |x|σ up, x ∈ \{0}, where ⊂RN(N≥3)is a bounded domain with 0 ∈, and p>1, σ>−2are given constants. For the case λ ≤(N−2)2/4, the singular behavior of all the positive solutions is completely classified in the recent paper [5]. Here we determine the exact singular behavior of all the positive solutions for the remaining case λ >(N−2)2/4. In sharp contrast to the case λ ≤(N−2)2/4, where several converging/blow-up rates of u(x)are possible as |x| →0, we show that when λ >(N−2)2/4, every positive solution u(x)blows up in the same fashion: lim |x|→0 |x| 2+σ p−1 u(x) = λ+ 2 +σ p −1 2 +σ p − 1 +2 −N 1/(p−1) .

We study the existence of bubbling solutions for the the following Chern-Simons-Higgs equation...

We study the behavior of finite Morse index solutions of the equation ... We show that the behavior depends crucially on whether p is below or above the critical power p(α). We also demonstrate how a duality method can be used to obtain sharper results.

We study the behavior of finite Morse index solutions to the weighted elliptic equation...

By introducing a suitable setting, we study the behavior of finite Morse-index solutions of the equation... We show that under our chosen setting for the finite Morse-index theory of (1), the stability of a solution to (1) is unchanged under various natural transformations. This enables us to reveal two critical values of the exponent p in (1) that divide the behavior of finite Morse-index solutions of (1), which in turn yields two critical powers for (2) through the transformation. The latter appear difficult to obtain by working directly with (2).

In earlier work, we provided a Hilbert manifold structure for the phase space for the Einstein-Yang-Mills equations, and used this to prove a condition for initial data to be stationary [S. McCormick, Adv. Theor. Math. Phys. 18, 799 (2014)]. Here we use the same phase space to consider the evolution of initial data exterior to some closed 2-surface boundary, and establish a condition for stationarity in this case. It is shown that the differential relationship given in the first law of black hole mechanics is exactly the condition required for the initial data to be stationary; this was first argued nonrigorously by Sudarsky and Wald [Phys. Rev. D 46, 1453 (1992)]. Furthermore, we give evidence to suggest that if this differential relationship holds then the boundary surface is the bifurcation surface of a bifurcate Killing horizon.

In this article, we study the dynamical behaviour of a new species spreading from a location in a river network where two or three branches meet, based on the widely used Fisher-KPP advection-diffusion equation. This local river system is represented by some simple graphs with every edge a half infinite line, meeting at a single vertex. We obtain a rather complete description of the long-time dynamical behaviour for every case under consideration, which can be classified into three different types (called a trichotomy), according to the water flow speeds in the river branches, which depend crucially on the topological structure of the graph representing the local river system and on the cross section areas of the branches. The trichotomy includes two different kinds of persistence states, and the state called "persistence below carrying capacity" here appears new.

We consider a reaction-diffusion-advection equation of the form: ut = uxx - β(t)ux + f (t, u) for x ∈ (g(t), h(t)), where β(t) is a T-periodic function representing the intensity of the advection, f (t, u) is a Fisher-KPP type of nonlinearity, T periodic in t, g(t) and h(t) are two free boundaries satisfying Stefan conditions. This equation can be used to describe the population dynamics in time-periodic environment with advection. Its homogeneous version (that is, both β and f are independent of t) was recently studied by Gu et al. (J Funct Anal 269:1714-1768, 2015). In this paper we consider the time-periodic case and study the long time behavior of the solutions. We show that a vanishing-spreading dichotomy result holds when β is small; a vanishing transition-virtual spreading trichotomy result holds when β is a medium-sized function; all solutions vanish when β is large. Here the partition of β(t) depends not only on the "size" β := 1T ∫ T0 β(t)dt of β(t) but also on its "shape" β(t):=β(t)-β.

In [2] and [3], fountain theorems and their dual forms in Banach space were established respectively. They are effective tools in studying the existence of infinitely many solutions. It should be noted that a decomposition of the Banach space plays an important role in proving these theorems. The decomposition allows one to apply Borsuk-Ulam theorem to establish a proper intersection lemma. Such a decomposition in many cases is done by using the eigenspaces of operators concerned. However, there are many operators, for instance, the p-Laplacian operator -Δp, whose spectrum are not very well understood. In recent works [5], [6], [7], a linking theorem over cones was obtained, and solutions for a quasilinear elliptic problem were found. In the use of the theorem, it does not require a complete decomposition of spaces. In this paper, we first establish a fountain theorem over cones in Banach spaces.

As vectors, mosquitoes transmit numerous mosquito-borne diseases. Among the many factors affecting the distribution and density of mosquitoes, climate change and warming have been increasingly recognized as major ones. In this paper, we make use of three diffusive logistic models with free boundary in one space dimension to explore the impact of climate warming on the movement of mosquito range. First, a general model incorporating temperature change with location and time is introduced. In order to gain insights of the model, a simplified version of the model with the change of temperature depending only on location is analyzed theoretically, for which the dynamical behavior is completely determined and presented. The general model can be modified into a more realistic one of seasonal succession type, to take into account of the seasonal changes of mosquito movements during each year, where the general model applies only for the time period of the warm seasons of the year, and during the cold season, the mosquito range is fixed and the population is assumed to be in a hibernating status. For both the general model and the seasonal succession model, our numerical simulations indicate that the long-time dynamical behavior is qualitatively similar to the simplified model, and the effect of climate warming on the movement of mosquitoes can be easily captured. Moreover, our analysis reveals that hibernating enhances the chances of survival and successful spreading of the mosquitoes, but it slows down the spreading speed.

We consider the following Hénon equation with critical growth... Furthermore, we prove that equation (*) has infinitely many nonradial solutions, whose energy can be made arbitrarily large.

We consider the following nonlinear field equation with super-critical growth: (∗) −Δu + λu = Q(y)u(N+2)/(N−2), u>0 inRN+m, u(y) →0 as |y| → +∞, where m 1, λ 0 and Q(y) is a bounded positive function. We show that equation (*) has infinitely many positive solutions under certain symmetry conditions on Q(y).

In this paper, we explore the role of lower-dimensional Sobolev exponents on the existence and multiplicity of solutions to (1.1). Namely we consider the following equation with supercritical growth...

In this paper, simulated by the paper of Wei and Yan [33] (see also [30–32]), we intend to find infinitely many positive solutions to (1.2) for all p ∈ (1, 5) under weaker integrability conditions on K(y) and Q(y). In [33], a single equation, this is, K(y) ≡ 0 in (1.2), was studied and infinitely many non-radial solutions were found in the case that Q(y) is radial. For this, they employed a very novel idea, that is, they use k, the number of the bumps of the solutions, as the parameter to construct Infinitely Many Solutions for Schrödinger-Poisson System 1071 spike solutions for the Schrödinger equation considered.

The aim of this paper is to obtain infinitely many non-radial positive solutions for (1.7) under an assumption for V(r) near the infinity.