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1534-0392
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1553-5258
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Communications on Pure & Applied Analysis
December 2004 , Volume 3 , Issue 4
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2004, 3(4): 545-556
doi: 10.3934/cpaa.2004.3.545
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Abstract:
We study stability of subsonic phase boundary solutions in the Suliciu model for phase transitions under tri-linear structural relation. With the help of Laplace transform, the evolution of perturbation is described by a linear dynamical system, and explicit solution is obtained in terms of inverse Laplace transform. Stability is established through energy estimates. The relaxed system is also discussed.
We study stability of subsonic phase boundary solutions in the Suliciu model for phase transitions under tri-linear structural relation. With the help of Laplace transform, the evolution of perturbation is described by a linear dynamical system, and explicit solution is obtained in terms of inverse Laplace transform. Stability is established through energy estimates. The relaxed system is also discussed.
2004, 3(4): 557-580
doi: 10.3934/cpaa.2004.3.557
+[Abstract](2103)
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Abstract:
In this paper we study the problem $(P_{\varepsilon}): \Delta^2 u_\varepsilon = u_\varepsilon^{\frac{n+4}{n-4}}$, $ u_\varepsilon >0$ in $A_\varepsilon$; $u_\varepsilon= \Delta u_\varepsilon= 0$ on $ \partial A_\varepsilon $, where $\{A_\varepsilon \subset \mathbb R^n, \varepsilon >0 \}$ is a family of bounded annulus shaped domains such that $A_\varepsilon$ becomes "thin" as $\varepsilon\to 0$. Our main result is the following: Assume $n\geq 6$ and let $C>0$ be a constant. Then there exists $\varepsilon_0>0$ such that for any $\varepsilon <\varepsilon_0$, the problem $ (P_{\varepsilon})$ has no solution $u_\varepsilon$, whose energy, $\int_{A_\varepsilon}|\Delta u_\varepsilon |^2$ is less than $C$. Our proof involves a rather delicate analysis of asymptotic profiles of solutions $u_\varepsilon$ when $\varepsilon\to 0$.
In this paper we study the problem $(P_{\varepsilon}): \Delta^2 u_\varepsilon = u_\varepsilon^{\frac{n+4}{n-4}}$, $ u_\varepsilon >0$ in $A_\varepsilon$; $u_\varepsilon= \Delta u_\varepsilon= 0$ on $ \partial A_\varepsilon $, where $\{A_\varepsilon \subset \mathbb R^n, \varepsilon >0 \}$ is a family of bounded annulus shaped domains such that $A_\varepsilon$ becomes "thin" as $\varepsilon\to 0$. Our main result is the following: Assume $n\geq 6$ and let $C>0$ be a constant. Then there exists $\varepsilon_0>0$ such that for any $\varepsilon <\varepsilon_0$, the problem $ (P_{\varepsilon})$ has no solution $u_\varepsilon$, whose energy, $\int_{A_\varepsilon}|\Delta u_\varepsilon |^2$ is less than $C$. Our proof involves a rather delicate analysis of asymptotic profiles of solutions $u_\varepsilon$ when $\varepsilon\to 0$.
2004, 3(4): 581-606
doi: 10.3934/cpaa.2004.3.581
+[Abstract](1670)
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Abstract:
We derive an Ornstein-Zernike asymptotic formula for the decay of the two point finite connectivity function $\phi^f_p(x\leftrightarrow y)$ of the Bernoulli bond percolation process on $\mathbb Z^d$, in the supercritical phase, along the principal directions, for $d\ge 3$, and for values of $p$ sufficiently near to $p=1$.
We derive an Ornstein-Zernike asymptotic formula for the decay of the two point finite connectivity function $\phi^f_p(x\leftrightarrow y)$ of the Bernoulli bond percolation process on $\mathbb Z^d$, in the supercritical phase, along the principal directions, for $d\ge 3$, and for values of $p$ sufficiently near to $p=1$.
2004, 3(4): 607-635
doi: 10.3934/cpaa.2004.3.607
+[Abstract](2453)
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Abstract:
Motivated by physical models and the so-called Crocco equation, we study the controllability properties of a class of degenerate parabolic equations. Due to degeneracy, classical null controllability results do not hold for this problem in general.
First, we prove that we can drive the solution to rest at time $T$ in a suitable subset of the space domain (regional null controllability). However, unlike for nondegenerate parabolic equations, this property is no more automatically preserved with time. Then, we prove that, given a time interval $(T,T')$, we can control the equation up to $T'$ and remain at rest during all the time interval $(T,T')$ on the same subset of the space domain (persistent regional null controllability). The proofs of these results are obtained via new observability inequalities derived from classical Carleman estimates by an appropriate use of cut-off functions.
With the same method, we also derive results of regional controllability for a Crocco type linearized equation and for the nondegenerate heat equation in unbounded domains.
Motivated by physical models and the so-called Crocco equation, we study the controllability properties of a class of degenerate parabolic equations. Due to degeneracy, classical null controllability results do not hold for this problem in general.
First, we prove that we can drive the solution to rest at time $T$ in a suitable subset of the space domain (regional null controllability). However, unlike for nondegenerate parabolic equations, this property is no more automatically preserved with time. Then, we prove that, given a time interval $(T,T')$, we can control the equation up to $T'$ and remain at rest during all the time interval $(T,T')$ on the same subset of the space domain (persistent regional null controllability). The proofs of these results are obtained via new observability inequalities derived from classical Carleman estimates by an appropriate use of cut-off functions.
With the same method, we also derive results of regional controllability for a Crocco type linearized equation and for the nondegenerate heat equation in unbounded domains.
2004, 3(4): 637-651
doi: 10.3934/cpaa.2004.3.637
+[Abstract](2097)
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Abstract:
In this work we prove that global attractors of systems of weakly coupled parabolic equations with nonlinear boundary conditions and large diffusivity are close to attractors of an ordinary differential equation. The limiting ordinary differential equation is given explicitly in terms of the reaction, boundary flux, the $n$-dimensional Lebesgue measure of the domain and the $(n-1)-$Hausdorff measure of its boundary. The tools are invariant manifold theory and comparison results.
In this work we prove that global attractors of systems of weakly coupled parabolic equations with nonlinear boundary conditions and large diffusivity are close to attractors of an ordinary differential equation. The limiting ordinary differential equation is given explicitly in terms of the reaction, boundary flux, the $n$-dimensional Lebesgue measure of the domain and the $(n-1)-$Hausdorff measure of its boundary. The tools are invariant manifold theory and comparison results.
2004, 3(4): 653-662
doi: 10.3934/cpaa.2004.3.653
+[Abstract](3263)
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Abstract:
We consider the elliptic problems $\Delta u=a(x)u^m$, $m>1$, and $\Delta u=a(x)e^u$ in a smooth bounded domain $\Omega$, with the boundary condition $u=+\infty$ on $\partial\Omega$. The weight function $a(x)$ is assumed to be Hölder continuous, growing like a negative power of $d(x)=$ dist $(x,\partial\Omega)$ near $\partial\Omega$. We show existence and nonexistence results, uniqueness and asymptotic estimates near the boundary for both the solutions and their normal derivatives.
We consider the elliptic problems $\Delta u=a(x)u^m$, $m>1$, and $\Delta u=a(x)e^u$ in a smooth bounded domain $\Omega$, with the boundary condition $u=+\infty$ on $\partial\Omega$. The weight function $a(x)$ is assumed to be Hölder continuous, growing like a negative power of $d(x)=$ dist $(x,\partial\Omega)$ near $\partial\Omega$. We show existence and nonexistence results, uniqueness and asymptotic estimates near the boundary for both the solutions and their normal derivatives.
2004, 3(4): 663-674
doi: 10.3934/cpaa.2004.3.663
+[Abstract](1947)
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Abstract:
We prove existence of Steiner symmetric maximizers for a constrained variational problem in $\mathbb R^2$. Solutions represent steady geophysical flows over a surface of variable height. The kinetic energy is maximized with respect to the set formed by intersecting a set of rearrangements of a given function with an affine subspace of codimension one.
We prove existence of Steiner symmetric maximizers for a constrained variational problem in $\mathbb R^2$. Solutions represent steady geophysical flows over a surface of variable height. The kinetic energy is maximized with respect to the set formed by intersecting a set of rearrangements of a given function with an affine subspace of codimension one.
2004, 3(4): 675-694
doi: 10.3934/cpaa.2004.3.675
+[Abstract](2692)
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Abstract:
In this paper, we consider the one-dimensional compressible Navier-Stokes equations for isentropic flow connecting to vacuum state with a continuous density when viscosity coefficient depends on the density. Precisely, the viscosity coefficient $\mu$ is proportional to $\rho^\theta$ and $0<\theta<1/2$, where $\rho$ is the density. The global existence of weak solutions is proved.
In this paper, we consider the one-dimensional compressible Navier-Stokes equations for isentropic flow connecting to vacuum state with a continuous density when viscosity coefficient depends on the density. Precisely, the viscosity coefficient $\mu$ is proportional to $\rho^\theta$ and $0<\theta<1/2$, where $\rho$ is the density. The global existence of weak solutions is proved.
2004, 3(4): 695-727
doi: 10.3934/cpaa.2004.3.695
+[Abstract](2167)
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Abstract:
In this paper we investigate compactness properties for a semiflow generated by a semi-linear equation with non-dense domain. We start with the non-homogeneous linear case, and we derive some abstract conditions for non-autonomous semilinear equations. Then we investigate a special situation which is well adapted for age-structured equations. We conclude the paper by applying the abstract results to an age-structured model with an additional structure.
In this paper we investigate compactness properties for a semiflow generated by a semi-linear equation with non-dense domain. We start with the non-homogeneous linear case, and we derive some abstract conditions for non-autonomous semilinear equations. Then we investigate a special situation which is well adapted for age-structured equations. We conclude the paper by applying the abstract results to an age-structured model with an additional structure.
2004, 3(4): 729-756
doi: 10.3934/cpaa.2004.3.729
+[Abstract](2672)
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Abstract:
We study nonlinear Dirichlet problems driven by the scalar $p$-Laplacian with a nonsmooth potential. First for the so-called "sublinear problem", under nonuniform nonresonance conditions, we establish the existence of at least one strictly positive solution. Then we prove two multiplicity results for positive solutions. The first concerns the "superlinear problem" and the second is for the sublinear problem. The method of proof is variational based on the nonsmooth critical point theory for locally Lipschitz functions. Our results complement the ones obtained by De Coster (Nonlin.Anal.23 (1995)).
We study nonlinear Dirichlet problems driven by the scalar $p$-Laplacian with a nonsmooth potential. First for the so-called "sublinear problem", under nonuniform nonresonance conditions, we establish the existence of at least one strictly positive solution. Then we prove two multiplicity results for positive solutions. The first concerns the "superlinear problem" and the second is for the sublinear problem. The method of proof is variational based on the nonsmooth critical point theory for locally Lipschitz functions. Our results complement the ones obtained by De Coster (Nonlin.Anal.23 (1995)).
2004, 3(4): 757-774
doi: 10.3934/cpaa.2004.3.757
+[Abstract](2074)
+[PDF](195.3KB)
Abstract:
We prove semiconcavity of the value function of a nonlinear optimal control problem where the cost functional depends on the arrival time of the trajectory on a given target set. We make suitable smoothness assumptions on the dynamics of the system, while the target set can be completely general. As a corollary, we prove differentiability of the value function for a class of linear systems.
We prove semiconcavity of the value function of a nonlinear optimal control problem where the cost functional depends on the arrival time of the trajectory on a given target set. We make suitable smoothness assumptions on the dynamics of the system, while the target set can be completely general. As a corollary, we prove differentiability of the value function for a class of linear systems.
2004, 3(4): 775-790
doi: 10.3934/cpaa.2004.3.775
+[Abstract](2100)
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Abstract:
The equations for viscous, compressible, heat-conductive, real reactive flows in dynamic combustion are considered, where the equations of state are nonlinear in temperature unlike the linear dependence for perfect gases. The initial-boundary value problem with Dirichlet-Neumann mixed boundaries in a finite domain is studied. The existence, uniqueness, and regularity of global solutions are established with general large initial data in $H^1$. It is proved that, although the solutions have large oscillations, there is no shock wave, turbulence, vacuum, mass concentration, or extremely hot spot developed in any finite time.
The equations for viscous, compressible, heat-conductive, real reactive flows in dynamic combustion are considered, where the equations of state are nonlinear in temperature unlike the linear dependence for perfect gases. The initial-boundary value problem with Dirichlet-Neumann mixed boundaries in a finite domain is studied. The existence, uniqueness, and regularity of global solutions are established with general large initial data in $H^1$. It is proved that, although the solutions have large oscillations, there is no shock wave, turbulence, vacuum, mass concentration, or extremely hot spot developed in any finite time.
2004, 3(4): 791-808
doi: 10.3934/cpaa.2004.3.791
+[Abstract](4326)
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Abstract:
Weak and strong convergence for some generalized proximal point algorithms are proved. These algorithms include the Eckstein and Bertsekas generalized proximal point algorithm, a contraction-proximal point algorithm, and inexact proximal point algorithms. Convergence rate is also considered.
Weak and strong convergence for some generalized proximal point algorithms are proved. These algorithms include the Eckstein and Bertsekas generalized proximal point algorithm, a contraction-proximal point algorithm, and inexact proximal point algorithms. Convergence rate is also considered.
2004, 3(4): 809-848
doi: 10.3934/cpaa.2004.3.809
+[Abstract](2006)
+[PDF](680.9KB)
Abstract:
In this paper, we develop and analyze a new model describing electrorheological fluid flow. In contrast to existing models, which assume the electric field to be perpendicular to the velocity field and are thus restricted to simple shear flow and flows close to it, we consider the fluid as anisotropic and introduce a general constitutive relation based on a viscosity function that depends on the shear rate, the electric field strength, and on the angle between the electric and the velocity vectors. We study general flow problems under nonhomogeneous mixed boundary conditions with given values of velocities and surface forces on different parts of the boundary. We investigate both the case where the viscosity function is continuous and the case where it is singular for vanishing shear rate. In the latter case, the problem reduces to a variational inequality. Using methods of nonlinear analysis such as fixed point theory, monotonicity, and compactness, we establish existence results for the problems under consideration. Some efficient methods for the numerical solution of the problems are presented, and numerical results for the simulation of the fluid flow in electrorheological shock absorbers are given.
In this paper, we develop and analyze a new model describing electrorheological fluid flow. In contrast to existing models, which assume the electric field to be perpendicular to the velocity field and are thus restricted to simple shear flow and flows close to it, we consider the fluid as anisotropic and introduce a general constitutive relation based on a viscosity function that depends on the shear rate, the electric field strength, and on the angle between the electric and the velocity vectors. We study general flow problems under nonhomogeneous mixed boundary conditions with given values of velocities and surface forces on different parts of the boundary. We investigate both the case where the viscosity function is continuous and the case where it is singular for vanishing shear rate. In the latter case, the problem reduces to a variational inequality. Using methods of nonlinear analysis such as fixed point theory, monotonicity, and compactness, we establish existence results for the problems under consideration. Some efficient methods for the numerical solution of the problems are presented, and numerical results for the simulation of the fluid flow in electrorheological shock absorbers are given.
2004, 3(4): 849-881
doi: 10.3934/cpaa.2004.3.849
+[Abstract](3149)
+[PDF](315.2KB)
Abstract:
We consider a parabolic equation nonlinearly coupled with a damped semilinear wave equation. This system describes the evolution of the relative temperature $\vartheta$ and of the order parameter $\chi$ in a material subject to phase transitions in the framework of phase-field theories. The hyperbolic dynamics is characterized by the presence of the inertial term $\mu\partial_{t t}\chi$ with $\mu>0$. When $\mu=0$, we reduce to the well-known phase-field model of Caginalp type. The goal of the present paper is an asymptotic analysis from the viewpoint of infinite-dimensional dynamical systems. We first prove that the model, endowed with appropriate boundary conditions, generates a strongly continuous semigroup on a suitable phase-space $\mathcal V_0$, which possesses a universal attractor $\mathcal A_\mu$. Our main result establishes that $\mathcal A_\mu$ is bounded by a constant independent of $\mu$ in a smaller phase-space $\mathcal V_1$. This bound allows us to show that the lifting $\mathcal A_0$ of the universal attractor of the parabolic system (corresponding to $\mu=0$) is upper semicontinuous at $0$ with respect to the family $\{\mathcal A_\mu,\mu>0\}$. We also construct an exponential attractor; that is, a set of finite fractal dimension attracting all the trajectories exponentially fast with respect to the distance in $\mathcal V_0$. The existence of an exponential attractor is obtained in the case $\mu=0$ as well. Finally, a noteworthy consequence is that the above results also hold for the damped semilinear wave equation, which is obtained as a particular case of our system when the coupling term vanishes. This provides a generalization of a number of theorems proved in the last two decades.
We consider a parabolic equation nonlinearly coupled with a damped semilinear wave equation. This system describes the evolution of the relative temperature $\vartheta$ and of the order parameter $\chi$ in a material subject to phase transitions in the framework of phase-field theories. The hyperbolic dynamics is characterized by the presence of the inertial term $\mu\partial_{t t}\chi$ with $\mu>0$. When $\mu=0$, we reduce to the well-known phase-field model of Caginalp type. The goal of the present paper is an asymptotic analysis from the viewpoint of infinite-dimensional dynamical systems. We first prove that the model, endowed with appropriate boundary conditions, generates a strongly continuous semigroup on a suitable phase-space $\mathcal V_0$, which possesses a universal attractor $\mathcal A_\mu$. Our main result establishes that $\mathcal A_\mu$ is bounded by a constant independent of $\mu$ in a smaller phase-space $\mathcal V_1$. This bound allows us to show that the lifting $\mathcal A_0$ of the universal attractor of the parabolic system (corresponding to $\mu=0$) is upper semicontinuous at $0$ with respect to the family $\{\mathcal A_\mu,\mu>0\}$. We also construct an exponential attractor; that is, a set of finite fractal dimension attracting all the trajectories exponentially fast with respect to the distance in $\mathcal V_0$. The existence of an exponential attractor is obtained in the case $\mu=0$ as well. Finally, a noteworthy consequence is that the above results also hold for the damped semilinear wave equation, which is obtained as a particular case of our system when the coupling term vanishes. This provides a generalization of a number of theorems proved in the last two decades.
2004, 3(4): 883-919
doi: 10.3934/cpaa.2004.3.883
+[Abstract](2541)
+[PDF](349.4KB)
Abstract:
The notion of (homogeneous) exponential Besov spaces is introduced and the infinite smoothness of such spaces is shown. Moreover, we consider some applications of exponential Besov spaces to a class of evolution equations involving dissipative terms, such as Cauchy-Riemann equations, semi-linear parabolic equations and semi-linear viscoelastic equations. The existence, uniqueness and regularity of solutions for the Cauchy problem of these equations will be established with rough initial data.
The notion of (homogeneous) exponential Besov spaces is introduced and the infinite smoothness of such spaces is shown. Moreover, we consider some applications of exponential Besov spaces to a class of evolution equations involving dissipative terms, such as Cauchy-Riemann equations, semi-linear parabolic equations and semi-linear viscoelastic equations. The existence, uniqueness and regularity of solutions for the Cauchy problem of these equations will be established with rough initial data.
2004, 3(4): 921-934
doi: 10.3934/cpaa.2004.3.921
+[Abstract](3707)
+[PDF](176.4KB)
Abstract:
The paper is devoted to study of the longtime behavior of solutions of a damped semilinear wave equation in a bounded smooth domain of $\mathbb R^3$ with the nonautonomous external forces and with the critical cubic growth rate of the nonlinearity. In contrast to the previous papers, we prove the dissipativity of this equation in higher energy spaces $E^\alpha$, $0<\alpha\le 1$, without the usage of the dissipation integral (which is infinite in our case).
The paper is devoted to study of the longtime behavior of solutions of a damped semilinear wave equation in a bounded smooth domain of $\mathbb R^3$ with the nonautonomous external forces and with the critical cubic growth rate of the nonlinearity. In contrast to the previous papers, we prove the dissipativity of this equation in higher energy spaces $E^\alpha$, $0<\alpha\le 1$, without the usage of the dissipation integral (which is infinite in our case).
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