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Communications on Pure & Applied Analysis

2010 , Volume 9 , Issue 4

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Existence and uniqueness of a solution to a three-dimensional axially symmetric Biot problem arising in modeling blood flow
Taebeom Kim , Sunčica Čanić and  Giovanna Guidoboni
2010, 9(4): 839-865 doi: 10.3934/cpaa.2010.9.839 +[Abstract](50) +[PDF](324.1KB)
Abstract:
We prove the existence of a unique weak solution to a problem associated with studying blood flow in compliant, viscoelastic arteries. The model problem is a linearization of the leading-order approximation of a viscous, incompressible, Newtonian fluid flow in a long and slender viscoelastic tube with small aspect ratio. The resulting model is of Biot type. The linearized model equations form a hyperbolic-parabolic system of partial differential equations with degenerate diffusion. The degenerate diffusion is a consequence of the fact that the effects of the fluid viscosity in the axial direction of a long and slender tube are small in comparison with the effects of the fluid viscosity in the radial direction. Degenerate fluid diffusion and hyperbolicity of the hyperbolic-parabolic system cause lower regularity of a weak solution and are a source of the main difficulties associated with the existence proof. Crucial for the existence proof is the viscoelasticity of vessel walls which provides the main smoothing mechanisms in the energy estimates which, via the compactness arguments, leads to the proof of the existence of a solution of this problem. This has interesting consequences for the understanding of the underlying hemodynamics application. Our analysis shows that the viscoelasticity of the vessel walls is crucial in smoothing sharp wave fronts that might be generated by the steep pressure pulses emanating from the heart, which are known to occur in, for example, patients with aortic insufficiency.
Energy convexity estimates for non-degenerate ground states of nonlinear 1D Schrödinger systems
Eugenio Montefusco , Benedetta Pellacci and  Marco Squassina
2010, 9(4): 867-884 doi: 10.3934/cpaa.2010.9.867 +[Abstract](32) +[PDF](261.2KB)
Abstract:
We study the spectral structure of the complex linearized operator for a class of nonlinear Schrödinger systems, obtaining as byproduct some interesting properties of non-degenerate ground state of the associated elliptic system, such as being isolated and orbitally stable.
Weighted Sobolev embeddings and radial solutions of inhomogeneous quasilinear elliptic equations
Jiabao Su and  Rushun Tian
2010, 9(4): 885-904 doi: 10.3934/cpaa.2010.9.885 +[Abstract](38) +[PDF](268.1KB)
Abstract:
We study weighted Sobolev embeddings in radially symmetric function spaces and then investigate the existence of nontrivial radial solutions of inhomogeneous quasilinear elliptic equation with singular potentials and super-$(p, q)$-linear nonlinearity. The model equation is of the form

$ -\Delta_p u+V(|x|)|u|^{q-2}u=Q(|x|)|u|^{s-2}u, x\in R^N,$

$ u(x) \rightarrow 0,$ as $ |x|\rightarrow\infty. $

Finite difference approximation of the Mumford and Shah functional in a contact manifold of the Heisenberg space
Giovanna Citti , Maria Manfredini and  Alessandro Sarti
2010, 9(4): 905-927 doi: 10.3934/cpaa.2010.9.905 +[Abstract](42) +[PDF](314.0KB)
Abstract:
The functionality of the visual cortex has been described in [63] and in [50] as a contact manifold of dimension three and in [62] the Mumford and Shah functional has been proposed to segment lifting of an image in the three dimensional cortical space. Hence, we study here this functional and we provide a constructive approach to the problem, extending to the sub- Riemannian setting an approximation technique proposed by De Giorgi in the Euclidean case.
Nonlinear parabolic equations with a lower order term and $L^1$ data
Rosaria Di Nardo
2010, 9(4): 929-942 doi: 10.3934/cpaa.2010.9.929 +[Abstract](40) +[PDF](216.3KB)
Abstract:
In this paper we prove the existence of a renormalized solution for a class of nonlinear parabolic problems whose prototype is

$\frac{\partial u}{\partial t}-\Delta_p u+$ div $(c(x,t)|u|^{\gamma-1}u) =f $ in $Q_T$

$u(x,t)=0$ on $\partial\Omega\times(0,T) $

$u(x,0)=u_0 (x)$ in $\Omega,$

where $Q_T=\Omega\times(0,T),$ $\Omega$ is an open and bounded subset of $ \mathcal{R} ^N$, $N\geq2,$ $T>0,$ $\Delta_p$ is the so called $p$-Laplace operator, $\gamma=\frac{(N+2)(p-1)}{N+p},$ $c(x,t)\in(L^{\tau }(Q_{T}))^N,$ $\tau=\frac{N+p}{p-1},$ $\ f\in L^1 (Q_T), $ $u_{0}\in L^1(\Omega).$

Some properties of positive radial solutions for some semilinear elliptic equations
Zhuoran Du
2010, 9(4): 943-953 doi: 10.3934/cpaa.2010.9.943 +[Abstract](35) +[PDF](167.9KB)
Abstract:
We are interested in the singular elliptic equation

$ \Delta h = \frac{1}{\alpha } h^{ -\alpha }-p(r)$ in $R^N( N\geq3 ),$

where $ \alpha >1$ and the monotone decreasing function $p(r)$ satisfying $\lim_{r\rightarrow \infty}p(r)=c>0.$ In this paper we show that for any $ \eta >0 $ there is a unique radial solution $ h(r) $ with $ h(0)=\eta $ and $ h(r)$ is oscillatory in $ [0, \infty )$. We prove $ \lim_{r \rightarrow \infty } h(r)=( \alpha c)^{-\frac{1}{\alpha}}. $ We also obtain similar properties of singular solutions, of which the zero set is nonempty.

The Lin-Ni's conjecture for vector-valued Schrödinger equations in the closed case
Emmanuel Hebey
2010, 9(4): 955-962 doi: 10.3934/cpaa.2010.9.955 +[Abstract](37) +[PDF](168.6KB)
Abstract:
We prove that critical vector-valued Schrödinger equations on compact Riemannian manifolds possess only constant solutions when the potential is sufficiently small. We prove the result in dimension $n = 3$ for arbitrary manifolds and in dimension $n \ge 4$ for manifolds with positive curvature. We also establish a gap estimate on the smallness of the potentials for the specific case of $S^1(T)\times S^{n-1}$.
Translating solutions to mean curvature flow with a forcing term in Minkowski space
Hongjie Ju , Jian Lu and  Huaiyu Jian
2010, 9(4): 963-973 doi: 10.3934/cpaa.2010.9.963 +[Abstract](50) +[PDF](190.0KB)
Abstract:
We study the existence, uniqueness and asymptotic behavior of rotationally symmetric translating solutions to mean curvature flow with a forcing term in Minkowski space. As a result, a part of conjectures in [1] is proved.
Traveling fronts of curve flow with external force field
Huaiyu Jian , Hongjie Ju and  Wei Sun
2010, 9(4): 975-986 doi: 10.3934/cpaa.2010.9.975 +[Abstract](53) +[PDF](171.8KB)
Abstract:
This paper aims to classify all the traveling fronts of a curvature flow with external force fields in the two-dimensional Euclidean space, i.e., the curve is evolved by the sum of the curvature and an external force field. We show that any traveling front is either a line or Grim Reaper if the external force field is constant. However, we find that the traveling fronts are of completely different geometry for non-constant external force fields.
Coercive energy estimates for differential forms in semi-convex domains
Dorina Mitrea , Irina Mitrea , Marius Mitrea and  Lixin Yan
2010, 9(4): 987-1010 doi: 10.3934/cpaa.2010.9.987 +[Abstract](38) +[PDF](311.3KB)
Abstract:
In this paper, we prove a $H^1$-coercive estimate for differential forms of arbitrary degrees in semi-convex domains of the Euclidean space. The key result is an integral identity involving a boundary term in which the Weingarten matrix of the boundary intervenes, established for any Lipschitz domain $\Omega\subseteq \mathcal{R}^n$ whose outward unit normal $\nu$ belongs to $L^{n-1}_1(\partial\Omega)$, the $L^{n-1}$-based Sobolev space of order one on $\partial\Omega$.
Solutions for singular quasilinear Schrödinger equations with one parameter
João Marcos do Ó and  Abbas Moameni
2010, 9(4): 1011-1023 doi: 10.3934/cpaa.2010.9.1011 +[Abstract](38) +[PDF](176.6KB)
Abstract:
We establish the existence of positive bound state solutions for the singular quasilinear Schrödinger equation

$i\frac{\partial \psi}{\partial t}=- \Delta \psi+\psi + \bar{\omega} (|\psi |^2)\psi- \lambda \rho(|\psi|^2)\psi-\kappa\Delta \rho(|\psi|^2)\rho'(|\psi|^2)\psi, x \in \Omega,$

where $\bar{\omega} (\tau^2) \tau \rightarrow +\infty$ as $\tau \rightarrow 0$ and, $\lambda>0$ is a parameter and $\Omega$ is a ball in $\mathcal{R}^N$. This problem is studied in connection with the following quasilinear eigenvalue problem

$-\Delta \Psi-\kappa\Delta \rho(|\Psi|^2)\rho'(|\Psi|^2)\Psi =\lambda_1 \rho(|\Psi|^2)\Psi, x \in \Omega,$

Indeed, we establish the existence of solutions for the above Schrödinger equation when $\lambda$ belongs to a certain neighborhood of the first eigenvalue $\lambda_1$ of the above eigenvalue problem. The main feature of this paper is that the nonlinearity $ \bar{\omega} ( |\psi |^2)\psi$ is unbounded around the origin and also the presence of the second order nonlinear term. Our analysis shows the importance of the role played by the parameter $\lambda$ combined with the nonlinear nonhomogeneous term $-\Delta \rho(|\psi|^2)\rho'(|\psi|^2)\psi$. The proofs are based on various techniques related to variational methods and implicit function theorem.

Strongly nonlinear multivalued systems involving singular $\Phi$-Laplacian operators
Francesca Papalini
2010, 9(4): 1025-1040 doi: 10.3934/cpaa.2010.9.1025 +[Abstract](56) +[PDF](221.4KB)
Abstract:
In this paper we study two vector problems with homogeneous Dirichlet boundary conditions for second order strongly nonlinear differential inclusions involving a maximal monotone term. The first is governed by a nonlinear differential operator of the form $x\mapsto (k(t)\Phi(x'))'$, where $k\in C(T, R_+)$ and $\Phi$ is an increasing homeomorphism defined on a bounded domain. In this problem the maximal monotone term need not be defined everywhere in the state space $R^N$, incorporating into our framework differential variational inequalities. The second problem is governed by the more general differential operator of the type $x\mapsto (a(t,x)\Phi(x'))'$, where $a(t,x)$ is a positive and continuous scalar function. In this case the maximal monotone term is required to be defined everywhere.
Global solutions of the free boundary problem for the compressible Navier-Stokes equations with density-dependent viscosity
Xulong Qin and  Zheng-An Yao
2010, 9(4): 1041-1052 doi: 10.3934/cpaa.2010.9.1041 +[Abstract](52) +[PDF](185.9KB)
Abstract:
A free boundary problem is investigated for viscous, compressible, heat-conducting, one-dimensional real gas with general large initial data. More precisely, the viscosity is assumed to be $\mu(\rho)=\rho^{\lambda}(\lambda>0)$, where $\rho$ is the density of the gas, and there is nonlinear dependence upon the density and temperature for the equations of state which are different from the linear dependence of perfect gas. It is also proved that no shock wave, vacuum, mass or heat concentration will be developed in a finite time and that the free boundary (interface) separating the gas and vacuum expands at a finite velocity.
Solutions of a second-order Hamiltonian system with periodic boundary conditions
Qiong Meng and  X. H. Tang
2010, 9(4): 1053-1067 doi: 10.3934/cpaa.2010.9.1053 +[Abstract](40) +[PDF](175.7KB)
Abstract:
By using the least action principle and the minimax methods, some existence theorems are obtained existence of solutions to a second-order Hamiltonian system with periodic boundary conditions in the cases when the gradiant of the nonlinearity is bounded sublinearly and linearly respectively.
Periodic solutions of Hamiltonian systems with anisotropic growth
Tianqing An and  Zhi-Qiang Wang
2010, 9(4): 1069-1082 doi: 10.3934/cpaa.2010.9.1069 +[Abstract](51) +[PDF](174.7KB)
Abstract:
In this paper we obtain some existence and multiplicity results for periodic solutions of nonautonomous Hamiltonian systems $\dot z(t)=J\nabla H(z(t),t)$ whose Hamiltonian functions may have simultaneously, in different components, superquadratic, subquadratic and quadratic behaviors. Our results generalize some earlier work [3] of P. Felmer and [5] of P. Felmer and Z.-Q. Wang.
Continuous dependence in front propagation of convective reaction-diffusion equations
Luisa Malaguti , Cristina Marcelli and  Serena Matucci
2010, 9(4): 1083-1098 doi: 10.3934/cpaa.2010.9.1083 +[Abstract](48) +[PDF](229.1KB)
Abstract:
Continuous dependence of the threshold wave speed and of the travelling wave profiles for reaction-diffusion-convection equations

$ u_t + h(u)u_x = (d(u)u_x)_x + f(u)$

is here studied with respect to the diffusion, reaction and convection terms.

Regularity of the solution of a nonlinear wave equation
Út V. Lê
2010, 9(4): 1099-1115 doi: 10.3934/cpaa.2010.9.1099 +[Abstract](45) +[PDF](222.5KB)
Abstract:
The paper deals with the regularity of the weak solution of a nonlinear initial-boundary value problem given by a semi-linear wave equation with space-time dependent coefficients and a boundary-like antiperiodic condition.

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