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Networks and Heterogeneous Media
December 2012 , Volume 7 , Issue 4
Special Issue
dedicated to Hiroshi Matanoon the occasion of his 60th birthday: Part I
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2012, 7(4): i-iii
doi: 10.3934/nhm.2012.7.4i
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Abstract:
Professor Hiroshi Matano was born in Kyoto, Japan, on July 28th, 1952. He studied at Kyoto University, where he prepared his doctoral thesis under the supervision of Professor Masaya Yamaguti. He obtained his first academic position as a research associate at the University of Tokyo. He then moved to Hiroshima University in 1982 and came back to Tokyo in 1988. He has been a Professor at the Graduate School of Mathematical Sciences at the University of Tokyo since 1991.
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Professor Hiroshi Matano was born in Kyoto, Japan, on July 28th, 1952. He studied at Kyoto University, where he prepared his doctoral thesis under the supervision of Professor Masaya Yamaguti. He obtained his first academic position as a research associate at the University of Tokyo. He then moved to Hiroshima University in 1982 and came back to Tokyo in 1988. He has been a Professor at the Graduate School of Mathematical Sciences at the University of Tokyo since 1991.
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2012, 7(4): 583-603
doi: 10.3934/nhm.2012.7.583
+[Abstract](5806)
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Abstract:
We investigate, from a more ecological point of view, a free boundary model considered in [11] and [8] that describes the spreading of a new or invasive species, with the free boundary representing the spreading front. We derive the free boundary condition by considering a "population loss" at the spreading front, and correct some mistakes regarding the range of spreading speed in [11]. Then we use numerical simulation to gain further insights to the model, which may help to determine its usefulness in concrete ecological situations.
We investigate, from a more ecological point of view, a free boundary model considered in [11] and [8] that describes the spreading of a new or invasive species, with the free boundary representing the spreading front. We derive the free boundary condition by considering a "population loss" at the spreading front, and correct some mistakes regarding the range of spreading speed in [11]. Then we use numerical simulation to gain further insights to the model, which may help to determine its usefulness in concrete ecological situations.
2012, 7(4): 605-615
doi: 10.3934/nhm.2012.7.605
+[Abstract](3731)
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We study an optimal control problem for a semilinear elliptic boundary value problem giving rise to a free boundary. Here, the free boundary is generated due to the fact that the nonlinear term of the state equation is not differentiable. The new aspect considered in this paper, with respect to other control problems involving free boundaries, is that here the cost functional explicitly depends on the location of the free boundary. The main difficulty is to show the continuous dependence (in measure) of the free boundary with respect to the control function. The crucial tool to get such continuous dependence is to know how behaves the state solution near the free boundary, as in previous works by L.A. Caffarelli and D. Phillips among other authors. Here we improved previous results in the literature thanks to a suitable application of the Fleming-Rishel formula.
We study an optimal control problem for a semilinear elliptic boundary value problem giving rise to a free boundary. Here, the free boundary is generated due to the fact that the nonlinear term of the state equation is not differentiable. The new aspect considered in this paper, with respect to other control problems involving free boundaries, is that here the cost functional explicitly depends on the location of the free boundary. The main difficulty is to show the continuous dependence (in measure) of the free boundary with respect to the control function. The crucial tool to get such continuous dependence is to know how behaves the state solution near the free boundary, as in previous works by L.A. Caffarelli and D. Phillips among other authors. Here we improved previous results in the literature thanks to a suitable application of the Fleming-Rishel formula.
2012, 7(4): 617-659
doi: 10.3934/nhm.2012.7.617
+[Abstract](3084)
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Abstract:
We consider a semilinear parabolic equation of the form $u_t = u_{xx} + f(u,u_x)$ defined on the circle $x ∈ S^1=\mathbb{R}/2\pi\mathbb{Z}$. For a dissipative nonlinearity $f$ this equation generates a dissipative semiflow in the appropriate function space, and the corresponding global attractor $A_f$ is called a Sturm attractor. If $f=f(u,p)$ is even in $p$, then the semiflow possesses an embedded flow satisfying Neumann boundary conditions on the half-interval $(0,\pi)$. This is due to $O(2)$ equivariance of the semiflow and, more specifically, due to reflection at the axis through $x=0,\pi\in S^1$. For general $f=f(u,p)$, where only $SO(2)$ equivariance prevails, we will nevertheless use the Sturm permutation $\sigma$ introduced for the characterization of Neumann flows to obtain a purely combinatorial characterization of the Sturm attractors $A_f$ on the circle. With this Sturm permutation $\sigma$ we then enumerate and describe the heteroclinic connections of all Morse-Smale attractors $A_f$ with $m$ stationary solutions and $q$ periodic orbits, up to $n:=m+2q \le 9$.
We consider a semilinear parabolic equation of the form $u_t = u_{xx} + f(u,u_x)$ defined on the circle $x ∈ S^1=\mathbb{R}/2\pi\mathbb{Z}$. For a dissipative nonlinearity $f$ this equation generates a dissipative semiflow in the appropriate function space, and the corresponding global attractor $A_f$ is called a Sturm attractor. If $f=f(u,p)$ is even in $p$, then the semiflow possesses an embedded flow satisfying Neumann boundary conditions on the half-interval $(0,\pi)$. This is due to $O(2)$ equivariance of the semiflow and, more specifically, due to reflection at the axis through $x=0,\pi\in S^1$. For general $f=f(u,p)$, where only $SO(2)$ equivariance prevails, we will nevertheless use the Sturm permutation $\sigma$ introduced for the characterization of Neumann flows to obtain a purely combinatorial characterization of the Sturm attractors $A_f$ on the circle. With this Sturm permutation $\sigma$ we then enumerate and describe the heteroclinic connections of all Morse-Smale attractors $A_f$ with $m$ stationary solutions and $q$ periodic orbits, up to $n:=m+2q \le 9$.
2012, 7(4): 661-671
doi: 10.3934/nhm.2012.7.661
+[Abstract](3775)
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We present conjectures on asymptotic behaviour of threshold solutions of the Cauchy problem for a semilinear heat equation with Sobolev critical nonlinearity. The conjectures say that, depending on the decay rate of initial data and the space dimension, the threshold solutions may grow up, stabilize, or decay to zero as $t→∞$. The rates of grow up or decay are computed formally using matched asymptotics.
We present conjectures on asymptotic behaviour of threshold solutions of the Cauchy problem for a semilinear heat equation with Sobolev critical nonlinearity. The conjectures say that, depending on the decay rate of initial data and the space dimension, the threshold solutions may grow up, stabilize, or decay to zero as $t→∞$. The rates of grow up or decay are computed formally using matched asymptotics.
2012, 7(4): 673-689
doi: 10.3934/nhm.2012.7.673
+[Abstract](2852)
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We consider global bounded solutions of fully nonlinear parabolic equations on bounded reflectionally symmetric domains, under nonhomogeneous Dirichlet boundary condition. We assume that, as $t→∞$, the equation is asymptotically symmetric, the boundary condition is asymptotically homogeneous, and the solution is asymptotically strictly positive in the sense that all its limit profiles are strictly positive. Our main theorem states that all the limit profiles are reflectionally symmetric and decreasing on one side of the symmetry hyperplane in the direction perpendicular to the hyperplane. We also illustrate by example that, unlike for equations which are symmetric at all finite times, the result does not hold under a relaxed positivity condition requiring merely that at least one limit profile of the solution be strictly positive.
We consider global bounded solutions of fully nonlinear parabolic equations on bounded reflectionally symmetric domains, under nonhomogeneous Dirichlet boundary condition. We assume that, as $t→∞$, the equation is asymptotically symmetric, the boundary condition is asymptotically homogeneous, and the solution is asymptotically strictly positive in the sense that all its limit profiles are strictly positive. Our main theorem states that all the limit profiles are reflectionally symmetric and decreasing on one side of the symmetry hyperplane in the direction perpendicular to the hyperplane. We also illustrate by example that, unlike for equations which are symmetric at all finite times, the result does not hold under a relaxed positivity condition requiring merely that at least one limit profile of the solution be strictly positive.
2012, 7(4): 691-703
doi: 10.3934/nhm.2012.7.691
+[Abstract](5922)
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Abstract:
This article reviews biological processes that can be modeled by PDEs, it describes mathematical results, and suggests open problems. The first topic deals with tumor growth. This is modeled as a free boundary problem for a coupled system of elliptic, hyperbolic and parabolic equations. Existence theorems, stability of radially symmetric stationary solutions, and symmetry-breaking bifurcation results are stated. Next, a free boundary problem for wound healing is described, again involving a coupled system of PDEs. Other topics include movement of molecules in a neuron, modeled as a system of reaction-hyperbolic equations, and competition for resources, modeled as a system of reaction-diffusion equations.
This article reviews biological processes that can be modeled by PDEs, it describes mathematical results, and suggests open problems. The first topic deals with tumor growth. This is modeled as a free boundary problem for a coupled system of elliptic, hyperbolic and parabolic equations. Existence theorems, stability of radially symmetric stationary solutions, and symmetry-breaking bifurcation results are stated. Next, a free boundary problem for wound healing is described, again involving a coupled system of PDEs. Other topics include movement of molecules in a neuron, modeled as a system of reaction-hyperbolic equations, and competition for resources, modeled as a system of reaction-diffusion equations.
2012, 7(4): 705-740
doi: 10.3934/nhm.2012.7.705
+[Abstract](3758)
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In some species, one of the roles of pheromones is to influence aggregation behavior. We first propose a macroscopic cross-diffusion model for the self-organized aggregation of German cockroaches that includes directed movement due to an aggregation pheromone. We then propose a microscopic particle model which is set into context with the macroscopic model. Our goal is to link the macroscopic and microscopic descriptions by using the singular and the hydrodynamic limit procedures. A hybrid model related to the macroscopic and microscopic models is also proposed as a cockroach aggregation model. This hybrid model assumes that each individual responds to pheromone concentration and moves by two-mode simple symmetric random walks. It shows that even though the movement of individuals is not directed, two-mode simple symmetric random walks and effect of the pheromone result in self-organized aggregation.
In some species, one of the roles of pheromones is to influence aggregation behavior. We first propose a macroscopic cross-diffusion model for the self-organized aggregation of German cockroaches that includes directed movement due to an aggregation pheromone. We then propose a microscopic particle model which is set into context with the macroscopic model. Our goal is to link the macroscopic and microscopic descriptions by using the singular and the hydrodynamic limit procedures. A hybrid model related to the macroscopic and microscopic models is also proposed as a cockroach aggregation model. This hybrid model assumes that each individual responds to pheromone concentration and moves by two-mode simple symmetric random walks. It shows that even though the movement of individuals is not directed, two-mode simple symmetric random walks and effect of the pheromone result in self-organized aggregation.
2012, 7(4): 741-766
doi: 10.3934/nhm.2012.7.741
+[Abstract](3332)
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We consider the Navier-Stokes equations of an incompressible fluid in a three dimensional curved domain with permeable walls in the limit of small viscosity. Using a curvilinear coordinate system, adapted to the boundary, we construct a corrector function at order $ε^j$, $j=0,1$, where $ε$ is the (small) viscosity parameter. This allows us to obtain an asymptotic expansion of the Navier-Stokes solution at order $ε^j$, $j=0,1$, for $ε$ small . Using the asymptotic expansion, we prove that the Navier-Stokes solutions converge, as the viscosity parameter tends to zero, to the corresponding Euler solution in the natural energy norm. This work generalizes earlier results in [14] or [26], which discussed the case of a channel domain, while here the domain is curved.
We consider the Navier-Stokes equations of an incompressible fluid in a three dimensional curved domain with permeable walls in the limit of small viscosity. Using a curvilinear coordinate system, adapted to the boundary, we construct a corrector function at order $ε^j$, $j=0,1$, where $ε$ is the (small) viscosity parameter. This allows us to obtain an asymptotic expansion of the Navier-Stokes solution at order $ε^j$, $j=0,1$, for $ε$ small . Using the asymptotic expansion, we prove that the Navier-Stokes solutions converge, as the viscosity parameter tends to zero, to the corresponding Euler solution in the natural energy norm. This work generalizes earlier results in [14] or [26], which discussed the case of a channel domain, while here the domain is curved.
2012, 7(4): 767-780
doi: 10.3934/nhm.2012.7.767
+[Abstract](3484)
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This paper deals with the long-time behavior of solutions of nonlinear reaction-diffusion equations describing formation of morphogen gradients, the concentration fields of molecules acting as spatial regulators of cell differentiation in developing tissues. For the considered class of models, we establish existence of a new type of ultra-singular self-similar solutions. These solutions arise as limits of the solutions of the initial value problem with zero initial data and infinitely strong source at the boundary. We prove existence and uniqueness of such solutions in the suitable weighted energy spaces. Moreover, we prove that the obtained self-similar solutions are the long-time limits of the solutions of the initial value problem with zero initial data and a time-independent boundary source.
This paper deals with the long-time behavior of solutions of nonlinear reaction-diffusion equations describing formation of morphogen gradients, the concentration fields of molecules acting as spatial regulators of cell differentiation in developing tissues. For the considered class of models, we establish existence of a new type of ultra-singular self-similar solutions. These solutions arise as limits of the solutions of the initial value problem with zero initial data and infinitely strong source at the boundary. We prove existence and uniqueness of such solutions in the suitable weighted energy spaces. Moreover, we prove that the obtained self-similar solutions are the long-time limits of the solutions of the initial value problem with zero initial data and a time-independent boundary source.
2012, 7(4): 781-803
doi: 10.3934/nhm.2012.7.781
+[Abstract](2553)
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We consider a reaction-diffusion system of activator-inhibitor type arising in the theory of phase transition. It appears in biological contexts such as pattern formation in population genetics. The purpose of this work is to prove the convergence of the solution of this system to the solution of a free boundary Problem involving a motion by mean curvature.
We consider a reaction-diffusion system of activator-inhibitor type arising in the theory of phase transition. It appears in biological contexts such as pattern formation in population genetics. The purpose of this work is to prove the convergence of the solution of this system to the solution of a free boundary Problem involving a motion by mean curvature.
2012, 7(4): 805-836
doi: 10.3934/nhm.2012.7.805
+[Abstract](2677)
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We consider integro-differential models describing the evolution of a population structured by a quantitative trait. Individuals interact competitively, creating a strong selection pressure on the population. On the other hand, mutations are assumed to be small. Following the formalism of [20], this creates concentration phenomena, typically consisting in a sum of Dirac masses slowly evolving in time. We propose a modification to those classical models that takes the effect of small populations into accounts and corrects some abnormal behaviours.
We consider integro-differential models describing the evolution of a population structured by a quantitative trait. Individuals interact competitively, creating a strong selection pressure on the population. On the other hand, mutations are assumed to be small. Following the formalism of [20], this creates concentration phenomena, typically consisting in a sum of Dirac masses slowly evolving in time. We propose a modification to those classical models that takes the effect of small populations into accounts and corrects some abnormal behaviours.
2012, 7(4): 837-855
doi: 10.3934/nhm.2012.7.837
+[Abstract](2699)
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An entire solution of the Allen-Cahn equation $\Delta u=f(u)$, where $f$ is an odd function and has exactly three zeros at $\pm 1$ and $0$, e.g. $f(u)=u(u^2-1)$, is called a $2k$-ended solution if its nodal set is asymptotic to $2k$ half lines, and if along each of these half lines the function $u$ looks (up to a multiplication by $-1$) like the one dimensional, odd, heteroclinic solution $H$, of $H''=f(H)$. In this paper we present some recent advances in the theory of the multiple-end solutions. We begin with the description of the moduli space of such solutions. Next we move on to study a special class of these solutions with just four ends. A special example is the saddle solutions $U$ whose nodal lines are precisely the straight lines $y=\pm x$. We describe the connected components of the moduli space of $4$-ended solutions. Finally we establish a uniqueness result which gives a complete classification of these solutions. It says that all $4$-ended solutions are continuous deformations of the saddle solution.
An entire solution of the Allen-Cahn equation $\Delta u=f(u)$, where $f$ is an odd function and has exactly three zeros at $\pm 1$ and $0$, e.g. $f(u)=u(u^2-1)$, is called a $2k$-ended solution if its nodal set is asymptotic to $2k$ half lines, and if along each of these half lines the function $u$ looks (up to a multiplication by $-1$) like the one dimensional, odd, heteroclinic solution $H$, of $H''=f(H)$. In this paper we present some recent advances in the theory of the multiple-end solutions. We begin with the description of the moduli space of such solutions. Next we move on to study a special class of these solutions with just four ends. A special example is the saddle solutions $U$ whose nodal lines are precisely the straight lines $y=\pm x$. We describe the connected components of the moduli space of $4$-ended solutions. Finally we establish a uniqueness result which gives a complete classification of these solutions. It says that all $4$-ended solutions are continuous deformations of the saddle solution.
2012, 7(4): 857-879
doi: 10.3934/nhm.2012.7.857
+[Abstract](4353)
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We study a two-point free boundary problem in a sector for a quasilinear parabolic equation. The boundary conditions are assumed to be spatially and temporally "self-similar" in a special way. We prove the existence, uniqueness and asymptotic stability of an expanding solution which is self-similar at discrete times. We also study the existence and uniqueness of a shrinking solution which is self-similar at discrete times.
We study a two-point free boundary problem in a sector for a quasilinear parabolic equation. The boundary conditions are assumed to be spatially and temporally "self-similar" in a special way. We prove the existence, uniqueness and asymptotic stability of an expanding solution which is self-similar at discrete times. We also study the existence and uniqueness of a shrinking solution which is self-similar at discrete times.
2012, 7(4): 881-891
doi: 10.3934/nhm.2012.7.881
+[Abstract](2586)
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We study the behavior of unbounded global orbits in a class of strongly monotone semiflows and give a criterion for the existence of orbits with periodic growth. We also prove the uniqueness and asymptotic stability of such orbits. We apply our results to a certain class of nonlinear parabolic equations including a weakly anisotropic curvature flow in a two-dimensional annulus and show the convergence of the solutions to a periodically growing solution which grows up in infinite time changing its profile time-periodically.
We study the behavior of unbounded global orbits in a class of strongly monotone semiflows and give a criterion for the existence of orbits with periodic growth. We also prove the uniqueness and asymptotic stability of such orbits. We apply our results to a certain class of nonlinear parabolic equations including a weakly anisotropic curvature flow in a two-dimensional annulus and show the convergence of the solutions to a periodically growing solution which grows up in infinite time changing its profile time-periodically.
2012, 7(4): 893-926
doi: 10.3934/nhm.2012.7.893
+[Abstract](2980)
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Oscillatory dynamics in a reaction-diffusion system with spatially nonlocal effect under Neumann boundary conditions is studied. The system provides triply degenerate points for two spatially non-uniform modes and uniform one (zero mode). We focus our attention on the 0:1:2-mode interaction in the reaction-diffusion system. Using a normal form on the center manifold, we seek the equilibria and study the stability of them. Moreover, Hopf bifurcation phenomena is studied for each equilibrium which has a Hopf instability point. The numerical results to the chaotic dynamics are also shown.
Oscillatory dynamics in a reaction-diffusion system with spatially nonlocal effect under Neumann boundary conditions is studied. The system provides triply degenerate points for two spatially non-uniform modes and uniform one (zero mode). We focus our attention on the 0:1:2-mode interaction in the reaction-diffusion system. Using a normal form on the center manifold, we seek the equilibria and study the stability of them. Moreover, Hopf bifurcation phenomena is studied for each equilibrium which has a Hopf instability point. The numerical results to the chaotic dynamics are also shown.
2012, 7(4): 927-939
doi: 10.3934/nhm.2012.7.927
+[Abstract](3107)
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The existence of periodic and spatially heteroclinic solutions is studied for a class of semilinear elliptic partial differential equations.
The existence of periodic and spatially heteroclinic solutions is studied for a class of semilinear elliptic partial differential equations.
2012, 7(4): 941-966
doi: 10.3934/nhm.2012.7.941
+[Abstract](3029)
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A class of quasilinear parabolic equations of forward-backward type $u_t=[\phi(u)]_{xx}$ in one space dimension is addressed, under assumptions on the nonlinear term $\phi$ which hold for a number of mathematical models in the theory of phase transitions. The notion of a three-phase solution to the Cauchy problem associated with the aforementioned equation is introduced. Then the time evolution of three-phase solutions is investigated, relying on a suitable entropy inequality satisfied by such a solution. In particular, it is proven that transitions between stable phases must satisfy certain admissibility conditions.
A class of quasilinear parabolic equations of forward-backward type $u_t=[\phi(u)]_{xx}$ in one space dimension is addressed, under assumptions on the nonlinear term $\phi$ which hold for a number of mathematical models in the theory of phase transitions. The notion of a three-phase solution to the Cauchy problem associated with the aforementioned equation is introduced. Then the time evolution of three-phase solutions is investigated, relying on a suitable entropy inequality satisfied by such a solution. In particular, it is proven that transitions between stable phases must satisfy certain admissibility conditions.
2012, 7(4): 967-988
doi: 10.3934/nhm.2012.7.967
+[Abstract](2722)
+[PDF](432.8KB)
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We study non-cooperative, multi-component elliptic Schrödinger systems arising in nonlinear optics and Bose-Einstein condensation phenomena. We here reconsider the more delicate case of systems of $m ≥ 3$ components. We prove a Liouville-type nonexistence theorem in space dimensions $n ≥3$ for homogeneous nonlinearities of degree $ p < n/(n-2)$, under the optimal assumption that the associated matrix is strictly copositive. This extends recent work of Tavares, Terracini, Verzini and Weth [22] and of Quittner and the author [17], where the results were limited to $n ≤ 2$ dimensions or to $m=2$ components. The proof of the Liouville theorem is done by combining and improving different arguments from [22] and [17], namely a feedback procedure based on Rellich-Pohozaev type identities and functional analytic inequalities on $S^{n-1}$, and suitable test-function arguments. We also consider a more general class of systems with gradient structure, for which our arguments show the triviality of solutions satisfying a suitable integral bound, a result which may be of independent interest.
We study non-cooperative, multi-component elliptic Schrödinger systems arising in nonlinear optics and Bose-Einstein condensation phenomena. We here reconsider the more delicate case of systems of $m ≥ 3$ components. We prove a Liouville-type nonexistence theorem in space dimensions $n ≥3$ for homogeneous nonlinearities of degree $ p < n/(n-2)$, under the optimal assumption that the associated matrix is strictly copositive. This extends recent work of Tavares, Terracini, Verzini and Weth [22] and of Quittner and the author [17], where the results were limited to $n ≤ 2$ dimensions or to $m=2$ components. The proof of the Liouville theorem is done by combining and improving different arguments from [22] and [17], namely a feedback procedure based on Rellich-Pohozaev type identities and functional analytic inequalities on $S^{n-1}$, and suitable test-function arguments. We also consider a more general class of systems with gradient structure, for which our arguments show the triviality of solutions satisfying a suitable integral bound, a result which may be of independent interest.
2012, 7(4): 989-1018
doi: 10.3934/nhm.2012.7.989
+[Abstract](2928)
+[PDF](569.0KB)
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We study a nonlinear stationary system of transport equations with specific boundary conditions describing the transport of solutes dissolved in a fluid circulating in a countercurrent tubular architecture, which constitutes a simplified model of a kidney nephron. We prove that for every Lipschitz and monotonic nonlinearity (which stems from active transport across the ascending limb), the dynamic system, a PDE which we study through contraction properties, relaxes toward the unique stationary state. A study of the linearized stationary operator enables us, using eigenelements, to further show that under certain conditions regarding the nonlinearity, the relaxation is exponential. We also describe a finite volume scheme which allows us to efficiently approach the numerical solution to the stationary system. Finally, we apply this numerical method to illustrate how the countercurrent arrangement of tubes enhances the axial concentration gradient, thereby favoring the production of highly concentrated urine.
We study a nonlinear stationary system of transport equations with specific boundary conditions describing the transport of solutes dissolved in a fluid circulating in a countercurrent tubular architecture, which constitutes a simplified model of a kidney nephron. We prove that for every Lipschitz and monotonic nonlinearity (which stems from active transport across the ascending limb), the dynamic system, a PDE which we study through contraction properties, relaxes toward the unique stationary state. A study of the linearized stationary operator enables us, using eigenelements, to further show that under certain conditions regarding the nonlinearity, the relaxation is exponential. We also describe a finite volume scheme which allows us to efficiently approach the numerical solution to the stationary system. Finally, we apply this numerical method to illustrate how the countercurrent arrangement of tubes enhances the axial concentration gradient, thereby favoring the production of highly concentrated urine.
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