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1534-0392
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Communications on Pure & Applied Analysis
July 2016 , Volume 15 , Issue 4
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2016, 15(4): 1077-1105
doi: 10.3934/cpaa.2016.15.1077
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Abstract:
We investigate the large time behavior of a conserved phase-field system that describes the phase separation in a material with viscosity effects. We prove a well-posedness result, the existence of the global attractor and its upper semicontinuity, when the heat capacity tends to zero. Then we prove the existence of inertial manifolds in one space dimension, and for the case of a rectangular domain in two space dimension. We also construct robust families of exponential attractors that converge in the sense of upper and lower semicontinuity to those of the viscous Cahn-Hilliard equation. Continuity properties of the intersection of the inertial manifolds with bounded absorbing sets are also proven. This work extends and improves some recent results proven by A. Bonfoh for both the conserved and non-conserved phase-field systems.
We investigate the large time behavior of a conserved phase-field system that describes the phase separation in a material with viscosity effects. We prove a well-posedness result, the existence of the global attractor and its upper semicontinuity, when the heat capacity tends to zero. Then we prove the existence of inertial manifolds in one space dimension, and for the case of a rectangular domain in two space dimension. We also construct robust families of exponential attractors that converge in the sense of upper and lower semicontinuity to those of the viscous Cahn-Hilliard equation. Continuity properties of the intersection of the inertial manifolds with bounded absorbing sets are also proven. This work extends and improves some recent results proven by A. Bonfoh for both the conserved and non-conserved phase-field systems.
2016, 15(4): 1107-1123
doi: 10.3934/cpaa.2016.15.1107
+[Abstract](703)
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Abstract:
We consider nonlinear, nonhomogeneous and noncoercive Neumann problems with a Carathéodory reaction which is either $(p-1)$-superlinear near $\pm\infty$ (without satisfying the usual in such cases Ambrosetti-Rabinowitz condition) or $(p-1)$-sublinear near $\pm\infty$. Using variational methods and Morse theory (critical groups) we prove two existence theorems.
We consider nonlinear, nonhomogeneous and noncoercive Neumann problems with a Carathéodory reaction which is either $(p-1)$-superlinear near $\pm\infty$ (without satisfying the usual in such cases Ambrosetti-Rabinowitz condition) or $(p-1)$-sublinear near $\pm\infty$. Using variational methods and Morse theory (critical groups) we prove two existence theorems.
2016, 15(4): 1125-1138
doi: 10.3934/cpaa.2016.15.1125
+[Abstract](775)
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Abstract:
This paper concerns the following nonlinear Schrödinger equations: \begin{eqnarray} \left\{ \begin{array}{ll} \displaystyle -\varepsilon^2\Delta u +V(x)u= |u|^{p_+-2}u^++|u|^{p_--2}u^-,\ x\in\mathbb{R}^N,\\ \lim\limits_{|x|\rightarrow\infty}u(x)=0, \\ \end{array} \right. \end{eqnarray} where $N\geq 3$ and $2 < p_{\pm} < \frac{2N}{N-2}$. We obtain nodal solutions for the above nonlinear Schrödinger equations with decaying and vanishing potential at infinity, i.e., $\lim\limits_{|x|\rightarrow\infty}V(x)=0$.
This paper concerns the following nonlinear Schrödinger equations: \begin{eqnarray} \left\{ \begin{array}{ll} \displaystyle -\varepsilon^2\Delta u +V(x)u= |u|^{p_+-2}u^++|u|^{p_--2}u^-,\ x\in\mathbb{R}^N,\\ \lim\limits_{|x|\rightarrow\infty}u(x)=0, \\ \end{array} \right. \end{eqnarray} where $N\geq 3$ and $2 < p_{\pm} < \frac{2N}{N-2}$. We obtain nodal solutions for the above nonlinear Schrödinger equations with decaying and vanishing potential at infinity, i.e., $\lim\limits_{|x|\rightarrow\infty}V(x)=0$.
2016, 15(4): 1139-1156
doi: 10.3934/cpaa.2016.15.1139
+[Abstract](750)
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Abstract:
In this paper, we are interested in the problem of existence and uniqueness of a solution which belongs to class $\mathbb{D}$ for a backward stochastic differential equation with two strictly separated continuous reflecting barriers in the case when the data are $\mathbb{L}^1$-integrable and with generator satisfying the Lipschitz property. The main idea is to use the notion of local solution to obtain the global one.
In this paper, we are interested in the problem of existence and uniqueness of a solution which belongs to class $\mathbb{D}$ for a backward stochastic differential equation with two strictly separated continuous reflecting barriers in the case when the data are $\mathbb{L}^1$-integrable and with generator satisfying the Lipschitz property. The main idea is to use the notion of local solution to obtain the global one.
2016, 15(4): 1157-1178
doi: 10.3934/cpaa.2016.15.1157
+[Abstract](948)
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In this article we give an extension of the $L^2-$theory of anisotropic singular perturbations for elliptic problems. We study a linear and some nonlinear problems involving $L^{p}$ data ($1 < p < 2$). Convergences in pseudo Sobolev spaces are proved for weak and entropy solutions, and rate of convergence is given in cylindrical domains.
In this article we give an extension of the $L^2-$theory of anisotropic singular perturbations for elliptic problems. We study a linear and some nonlinear problems involving $L^{p}$ data ($1 < p < 2$). Convergences in pseudo Sobolev spaces are proved for weak and entropy solutions, and rate of convergence is given in cylindrical domains.
2016, 15(4): 1179-1191
doi: 10.3934/cpaa.2016.15.1179
+[Abstract](714)
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This paper is concerned with the initial boundary value problem of certain 2D MHD-$\alpha$ equations without velocity viscosity over a bounded domain with smooth boundary. We show that the equations have a unique global smooth solution $(u,b)$ for $W^{4,p}\times H^4$ initial data and physical boundary condition.
This paper is concerned with the initial boundary value problem of certain 2D MHD-$\alpha$ equations without velocity viscosity over a bounded domain with smooth boundary. We show that the equations have a unique global smooth solution $(u,b)$ for $W^{4,p}\times H^4$ initial data and physical boundary condition.
2016, 15(4): 1193-1213
doi: 10.3934/cpaa.2016.15.1193
+[Abstract](893)
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The present paper is devoted to the study of transition fronts of nonlocal Fisher-KPP equations in time heterogeneous media. We first construct transition fronts with exact decaying rates as the space variable tends to infinity and with prescribed interface location functions, which are natural generalizations of front location functions in homogeneous media. Then, by the general results on space regularity of transition fronts of nonlocal evolution equations proven in the authors' earlier work ([25]), these transition fronts are continuously differentiable in space. We show that their space partial derivatives have exact decaying rates as the space variable tends to infinity. Finally, we study the asymptotic stability of transition fronts. It is shown that transition fronts attract those solutions whose initial data decays as fast as transition fronts near infinity and essentially above zero near negative infinity.
The present paper is devoted to the study of transition fronts of nonlocal Fisher-KPP equations in time heterogeneous media. We first construct transition fronts with exact decaying rates as the space variable tends to infinity and with prescribed interface location functions, which are natural generalizations of front location functions in homogeneous media. Then, by the general results on space regularity of transition fronts of nonlocal evolution equations proven in the authors' earlier work ([25]), these transition fronts are continuously differentiable in space. We show that their space partial derivatives have exact decaying rates as the space variable tends to infinity. Finally, we study the asymptotic stability of transition fronts. It is shown that transition fronts attract those solutions whose initial data decays as fast as transition fronts near infinity and essentially above zero near negative infinity.
2016, 15(4): 1215-1231
doi: 10.3934/cpaa.2016.15.1215
+[Abstract](796)
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Abstract:
In this paper, we are concerned with the existence of least energy solutions of nonlinear Schrödinger equations involving the half Laplacian \begin{eqnarray} (-\Delta)^{1/2}u(x)+\lambda V(x)u(x)=u(x)^{p-1}, u(x)\geq 0, \quad x\in R^N, \end{eqnarray} for sufficiently large $\lambda$, $2 < p < \frac{2N}{N-1}$ for $N \geq 2$. $V(x)$ is a real continuous function on $R^N$. Using variational methods we prove the existence of least energy solution $u(x)$ which localize near the potential well int$(V^{-1}(0))$ for $\lambda$ large. Moreover, if the zero sets int$(V^{-1}(0))$ of $V(x)$ include more than one isolated components, then $u_\lambda(x)$ will be trapped around all the isolated components. However, in Laplacian case, when the parameter $\lambda$ large, the corresponding least energy solution will be trapped around only one isolated component and become arbitrary small in other components of int$(V^{-1}(0))$. This is the essential difference with the Laplacian problems since the operator $(-\Delta)^{1/2}$ is nonlocal.
In this paper, we are concerned with the existence of least energy solutions of nonlinear Schrödinger equations involving the half Laplacian \begin{eqnarray} (-\Delta)^{1/2}u(x)+\lambda V(x)u(x)=u(x)^{p-1}, u(x)\geq 0, \quad x\in R^N, \end{eqnarray} for sufficiently large $\lambda$, $2 < p < \frac{2N}{N-1}$ for $N \geq 2$. $V(x)$ is a real continuous function on $R^N$. Using variational methods we prove the existence of least energy solution $u(x)$ which localize near the potential well int$(V^{-1}(0))$ for $\lambda$ large. Moreover, if the zero sets int$(V^{-1}(0))$ of $V(x)$ include more than one isolated components, then $u_\lambda(x)$ will be trapped around all the isolated components. However, in Laplacian case, when the parameter $\lambda$ large, the corresponding least energy solution will be trapped around only one isolated component and become arbitrary small in other components of int$(V^{-1}(0))$. This is the essential difference with the Laplacian problems since the operator $(-\Delta)^{1/2}$ is nonlocal.
2016, 15(4): 1233-1250
doi: 10.3934/cpaa.2016.15.1233
+[Abstract](733)
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We consider a class of nearly integrable Hamiltonian systems with Hamiltonian being $H(\theta,I,u,v)=h(I)+\frac{1}{2}\sum_{j=1}^{m}\Omega_j(u_j^2-v_j^2)+f(\theta,I,u,v)$. By introducing external parameter and KAM methods, we prove that, if the frequency mapping has nonzero Brouwer topological degree at some Diophantine frequency, the hyperbolic invariant torus with this frequency persists under small perturbations.
We consider a class of nearly integrable Hamiltonian systems with Hamiltonian being $H(\theta,I,u,v)=h(I)+\frac{1}{2}\sum_{j=1}^{m}\Omega_j(u_j^2-v_j^2)+f(\theta,I,u,v)$. By introducing external parameter and KAM methods, we prove that, if the frequency mapping has nonzero Brouwer topological degree at some Diophantine frequency, the hyperbolic invariant torus with this frequency persists under small perturbations.
2016, 15(4): 1251-1263
doi: 10.3934/cpaa.2016.15.1251
+[Abstract](778)
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We derive the long time asymptotic of solutions to an evolutive Hamilton-Jacobi-Bellman equation in a bounded smooth domain, in connection with ergodic problems recently studied in [1]. Our main assumption is an appropriate degeneracy condition on the operator at the boundary. This condition is related to the characteristic boundary points for linear operators as well as to the irrelevant points for the generalized Dirichlet problem, and implies in particular that no boundary datum has to be imposed. We prove that there exists a constant $c$ such that the solutions of the evolutive problem converge uniformly, in the reference frame moving with constant velocity $c$, to a unique steady state solving a suitable ergodic problem.
We derive the long time asymptotic of solutions to an evolutive Hamilton-Jacobi-Bellman equation in a bounded smooth domain, in connection with ergodic problems recently studied in [1]. Our main assumption is an appropriate degeneracy condition on the operator at the boundary. This condition is related to the characteristic boundary points for linear operators as well as to the irrelevant points for the generalized Dirichlet problem, and implies in particular that no boundary datum has to be imposed. We prove that there exists a constant $c$ such that the solutions of the evolutive problem converge uniformly, in the reference frame moving with constant velocity $c$, to a unique steady state solving a suitable ergodic problem.
2016, 15(4): 1265-1283
doi: 10.3934/cpaa.2016.15.1265
+[Abstract](865)
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In the present paper, we consider the initial value problem for semilinear damped wave equations in one space dimension. Wakasugi [7] have obtained an upper bound of the lifespan for the problem only in the subcritical case. On the other hand, D'Abbicco $\&$ Lucente $\&$ Reissig [3] showed a blow-up result in the critical case. The aim of this paper is to give an estimate of the upper bound of the lifespan in the critical case, and show the optimality of the upper bound. Also, we derive an estimate of the lower bound of the lifespan in the subcritical case which shows the optimality of the upper bound in [7]. Moreover, we show that the critical exponent changes when the initial data are odd functions.
In the present paper, we consider the initial value problem for semilinear damped wave equations in one space dimension. Wakasugi [7] have obtained an upper bound of the lifespan for the problem only in the subcritical case. On the other hand, D'Abbicco $\&$ Lucente $\&$ Reissig [3] showed a blow-up result in the critical case. The aim of this paper is to give an estimate of the upper bound of the lifespan in the critical case, and show the optimality of the upper bound. Also, we derive an estimate of the lower bound of the lifespan in the subcritical case which shows the optimality of the upper bound in [7]. Moreover, we show that the critical exponent changes when the initial data are odd functions.
2016, 15(4): 1285-1308
doi: 10.3934/cpaa.2016.15.1285
+[Abstract](995)
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We study the combined effect of concave and convex nonlinearities on the number of positive solutions for a fractional system involving critical Sobolev exponents. With the help of the Nehari manifold, we prove that the system admits at least two positive solutions when the pair of parameters $(\lambda,\mu)$ belongs to a suitable subset of $R^2$.
We study the combined effect of concave and convex nonlinearities on the number of positive solutions for a fractional system involving critical Sobolev exponents. With the help of the Nehari manifold, we prove that the system admits at least two positive solutions when the pair of parameters $(\lambda,\mu)$ belongs to a suitable subset of $R^2$.
2016, 15(4): 1309-1333
doi: 10.3934/cpaa.2016.15.1309
+[Abstract](926)
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This paper is concerned with the existence of soliton solutions for a quasilinear Schrödinger equation in $R^N$ with critical exponent, which appears from modelling the self-channeling of a high-power ultrashort laser in matter. By working with a perturbation approach which was initially proposed in [26], we prove that the given problem has a positive ground state solution.
This paper is concerned with the existence of soliton solutions for a quasilinear Schrödinger equation in $R^N$ with critical exponent, which appears from modelling the self-channeling of a high-power ultrashort laser in matter. By working with a perturbation approach which was initially proposed in [26], we prove that the given problem has a positive ground state solution.
2016, 15(4): 1335-1350
doi: 10.3934/cpaa.2016.15.1335
+[Abstract](850)
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In this paper, we study the Dirichlet problem arising in the electrorheological fluids \begin{eqnarray} \begin{cases} -{\rm div}\ a(x,Du)=k(u^{\gamma-1}-u^{\beta-1}) & x\in \Omega, \\ u=0 & x\in \partial \Omega, \end{cases} \end{eqnarray} where $\Omega$ is a bounded domain in $R^n$ and ${\rm div}\ a(x,Du)$ is a $p(x)$-Laplace type operator with $1<\beta<\gamma<\inf_{x\in \Omega} p(x)$, $p(x)\in(1,2]$. By establish a reversed Hölder inequality, we show that for any suitable $\gamma,\beta$, the weak solution of previous equation has bounded $p(x)$ energy satisfies $|Du|^{p(x)}\in L_{\text{loc}}^{\delta}$ with some $\delta>1$.
In this paper, we study the Dirichlet problem arising in the electrorheological fluids \begin{eqnarray} \begin{cases} -{\rm div}\ a(x,Du)=k(u^{\gamma-1}-u^{\beta-1}) & x\in \Omega, \\ u=0 & x\in \partial \Omega, \end{cases} \end{eqnarray} where $\Omega$ is a bounded domain in $R^n$ and ${\rm div}\ a(x,Du)$ is a $p(x)$-Laplace type operator with $1<\beta<\gamma<\inf_{x\in \Omega} p(x)$, $p(x)\in(1,2]$. By establish a reversed Hölder inequality, we show that for any suitable $\gamma,\beta$, the weak solution of previous equation has bounded $p(x)$ energy satisfies $|Du|^{p(x)}\in L_{\text{loc}}^{\delta}$ with some $\delta>1$.
2016, 15(4): 1351-1370
doi: 10.3934/cpaa.2016.15.1351
+[Abstract](996)
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In this paper, we consider the existence and multiplicity of sign-changing solutions to some Kirchhoff-type equation involving a nonlinear term with exponential growth. In a first result, we prove the existence of at least three solutions: one solution is positive, one is negative and the third one is sign-changing. The existence of infinitely many sign-changing solutions is proved as our second result in this work. Our method is mainly based on invariant sets of descending flow in the framework of classical critical point theory.
In this paper, we consider the existence and multiplicity of sign-changing solutions to some Kirchhoff-type equation involving a nonlinear term with exponential growth. In a first result, we prove the existence of at least three solutions: one solution is positive, one is negative and the third one is sign-changing. The existence of infinitely many sign-changing solutions is proved as our second result in this work. Our method is mainly based on invariant sets of descending flow in the framework of classical critical point theory.
2016, 15(4): 1371-1399
doi: 10.3934/cpaa.2016.15.1371
+[Abstract](705)
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We consider the plasma-vacuum interface problem in a classical statement when in the plasma region the flow is governed by the equations of ideal compressible magnetohydrodynamics, while in the vacuum region the magnetic field obeys the div-curl system of pre-Maxwell dynamics. The local-in-time existence and uniqueness of the solution to this problem in suitable anisotropic Sobolev spaces was proved in [17], provided that at each point of the initial interface the plasma density is strictly positive and the magnetic fields on either side of the interface are not collinear. The non-collinearity condition appears as the requirement that the symbol associated to the interface is elliptic. We now consider the case when this symbol is not elliptic and study the linearized problem, provided that the unperturbed plasma and vacuum non-zero magnetic fields are collinear on the interface. We prove a basic a priori $L^2$ estimate for this problem under the (generalized) Rayleigh-Taylor sign condition $[\partial q/\partial N]<0$ on the jump of the normal derivative of the unperturbed total pressure satisfied at each point of the interface. By constructing an Hadamard-type ill-posedness example for the frozen coefficients linearized problem we show that the simultaneous failure of the non-collinearity condition and the Rayleigh-Taylor sign condition leads to Rayleigh-Taylor instability.
We consider the plasma-vacuum interface problem in a classical statement when in the plasma region the flow is governed by the equations of ideal compressible magnetohydrodynamics, while in the vacuum region the magnetic field obeys the div-curl system of pre-Maxwell dynamics. The local-in-time existence and uniqueness of the solution to this problem in suitable anisotropic Sobolev spaces was proved in [17], provided that at each point of the initial interface the plasma density is strictly positive and the magnetic fields on either side of the interface are not collinear. The non-collinearity condition appears as the requirement that the symbol associated to the interface is elliptic. We now consider the case when this symbol is not elliptic and study the linearized problem, provided that the unperturbed plasma and vacuum non-zero magnetic fields are collinear on the interface. We prove a basic a priori $L^2$ estimate for this problem under the (generalized) Rayleigh-Taylor sign condition $[\partial q/\partial N]<0$ on the jump of the normal derivative of the unperturbed total pressure satisfied at each point of the interface. By constructing an Hadamard-type ill-posedness example for the frozen coefficients linearized problem we show that the simultaneous failure of the non-collinearity condition and the Rayleigh-Taylor sign condition leads to Rayleigh-Taylor instability.
2016, 15(4): 1401-1417
doi: 10.3934/cpaa.2016.15.1401
+[Abstract](742)
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We show a result of maximal regularity in spaces of Hölder continuous function, concerning linear parabolic systems, with dynamic or Wentzell boundary conditions, with an elliptic diffusion term on the boundary.
We show a result of maximal regularity in spaces of Hölder continuous function, concerning linear parabolic systems, with dynamic or Wentzell boundary conditions, with an elliptic diffusion term on the boundary.
2016, 15(4): 1419-1449
doi: 10.3934/cpaa.2016.15.1419
+[Abstract](1194)
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In the present article we study the viscous Cahn-Hilliard-Navier-Stokes model, endowed with dynamic boundary conditions, from the theoretical and numerical point of view. We start by deducing results on the existence, uniqueness and regularity of the solutions for the continuous problem. Then we propose a space semi-discrete finite element approximation of the model and we study the convergence of the approximate scheme. We also prove the stability and convergence of a fully discretized scheme, obtained using the semi-implicit Euler scheme applied to the space semi-discretization proposed previously. Numerical simulations are also presented to illustrate the theoretical results.
In the present article we study the viscous Cahn-Hilliard-Navier-Stokes model, endowed with dynamic boundary conditions, from the theoretical and numerical point of view. We start by deducing results on the existence, uniqueness and regularity of the solutions for the continuous problem. Then we propose a space semi-discrete finite element approximation of the model and we study the convergence of the approximate scheme. We also prove the stability and convergence of a fully discretized scheme, obtained using the semi-implicit Euler scheme applied to the space semi-discretization proposed previously. Numerical simulations are also presented to illustrate the theoretical results.
2016, 15(4): 1451-1469
doi: 10.3934/cpaa.2016.15.1451
+[Abstract](934)
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In reaction-diffusion models describing the interaction between the invading grey squirrel and the established red squirrel in Britain, Okubo et al. ([19]) found that in certain parameter regimes, the profiles of the two species in a wave propagation solution can be determined via a solution of the KPP equation. Motivated by their result, we employ an elementary approach based on the maximum principle for elliptic inequalities coupled with estimates of a total density of the three species to establish the nonexistence of traveling wave solutions for Lotka-Volterra systems of three competing species. Applying our estimates to the May-Leonard model, we obtain upper and lower bounds for the total density of a solution to this system. For the existence of traveling wave solutions to the Lotka-Volterra three-species competing system, we find new semi-exact solutions by virtue of functions other than hyperbolic tangent functions, which are used in constructing one-hump exact traveling wave solutions in [2]. Moreover, new two-hump semi-exact traveling wave solutions different from the ones found in [1] are constructed.
In reaction-diffusion models describing the interaction between the invading grey squirrel and the established red squirrel in Britain, Okubo et al. ([19]) found that in certain parameter regimes, the profiles of the two species in a wave propagation solution can be determined via a solution of the KPP equation. Motivated by their result, we employ an elementary approach based on the maximum principle for elliptic inequalities coupled with estimates of a total density of the three species to establish the nonexistence of traveling wave solutions for Lotka-Volterra systems of three competing species. Applying our estimates to the May-Leonard model, we obtain upper and lower bounds for the total density of a solution to this system. For the existence of traveling wave solutions to the Lotka-Volterra three-species competing system, we find new semi-exact solutions by virtue of functions other than hyperbolic tangent functions, which are used in constructing one-hump exact traveling wave solutions in [2]. Moreover, new two-hump semi-exact traveling wave solutions different from the ones found in [1] are constructed.
2016, 15(4): 1471-1495
doi: 10.3934/cpaa.2016.15.1471
+[Abstract](808)
+[PDF](625.1KB)
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A time-delayed reaction-diffusion system of mistletoes and birds with nonlocal effect in a two-dimensional strip is considered in this paper. By the background of model deriving, the bird diffuses with a Neumann boundary value condition, and the mistletoes does not diffuse and thus without boundary value condition. Making use of the theory of monotone semiflows and Kuratowski measure of non-compactness, we discuss the existence of spreading speed $c^\ast$. The value of $c^*$ is evaluated by using two auxiliary linear systems accompanied with approximate process.
A time-delayed reaction-diffusion system of mistletoes and birds with nonlocal effect in a two-dimensional strip is considered in this paper. By the background of model deriving, the bird diffuses with a Neumann boundary value condition, and the mistletoes does not diffuse and thus without boundary value condition. Making use of the theory of monotone semiflows and Kuratowski measure of non-compactness, we discuss the existence of spreading speed $c^\ast$. The value of $c^*$ is evaluated by using two auxiliary linear systems accompanied with approximate process.
2016, 15(4): 1497-1514
doi: 10.3934/cpaa.2016.15.1497
+[Abstract](843)
+[PDF](613.0KB)
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We study exact multiplicity and bifurcation curves of positive solutions of the boundary value problem \begin{eqnarray} &u"(x)+\lambda (-u^4+\sigma u^3-\tau u^2+\rho u)=0, -1 < x < 1, \\ &u(-1)=u(1)=0, \end{eqnarray} where $\sigma, \tau \in \mathbb{R}$, $\rho \geq 0,$ and $\lambda >0$ is a bifurcation parameter. Then on the $(\lambda, \|u\|_\infty)$-plane, we give a classification of four qualitatively different bifurcation curves: an S-shaped curve, a broken S-shaped curve, a $\subset$-shaped curve and a monotone increasing curve.
We study exact multiplicity and bifurcation curves of positive solutions of the boundary value problem \begin{eqnarray} &u"(x)+\lambda (-u^4+\sigma u^3-\tau u^2+\rho u)=0, -1 < x < 1, \\ &u(-1)=u(1)=0, \end{eqnarray} where $\sigma, \tau \in \mathbb{R}$, $\rho \geq 0,$ and $\lambda >0$ is a bifurcation parameter. Then on the $(\lambda, \|u\|_\infty)$-plane, we give a classification of four qualitatively different bifurcation curves: an S-shaped curve, a broken S-shaped curve, a $\subset$-shaped curve and a monotone increasing curve.
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