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Communications on Pure & Applied Analysis

2009 , Volume 8 , Issue 4

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Study of the cyclicity of some degenerate graphics inside quadratic systems
Freddy Dumortier and Christiane Rousseau
2009, 8(4): 1133-1157 doi: 10.3934/cpaa.2009.8.1133 +[Abstract](233) +[PDF](302.6KB)
Abstract:
In this paper we make essential steps in proving the finite cyclicity of degenerate graphics in quadratic systems, having a line of singular points in the finite plane. In particular we consider the graphics $(DF_{1 a})$, $(DF_{2 a})$ of the program of [8] to prove the finiteness part of Hilbert's 16th problem for quadratic vector fields. We make a complete treatment except for one very specific problem that we clearly identify.
On a class of hypoelliptic operators with unbounded coefficients in $R^N$
Bálint Farkas and Luca Lorenzi
2009, 8(4): 1159-1201 doi: 10.3934/cpaa.2009.8.1159 +[Abstract](251) +[PDF](514.4KB)
Abstract:
We consider second order linear partial differential operators $A$ on $R^N$ which are not assumed to be uniformly elliptic and whose coefficients in the second order part may grow quadratically, while the drift part has essentially linear growth. Instead of uniform ellipticity, we require a much weaker hypothesis of uniform hypoellipticity, which in an equivalent formulation connects the behaviour of the diffusion and the drift coefficients, by requiring that a Kalman-type condition is satisfied for them. By refining Bernstein's method we prove the existence of a semigroup {$T(t)$} of bounded linear operators (in the space of bounded and continuous functions) associated to the operator $A$. We also show uniform estimates for the spatial derivatives of the semigroup {$T(t)$} in (an)isotropic spaces of (Hölder-) continuous functions. As a consequence, we obtain Hölder estimates for the solutions of some elliptic and parabolic problems associated to the operator $A$.
Weak solutions of linear degenerate parabolic equations and an application in image processing
Kristian Bredies
2009, 8(4): 1203-1229 doi: 10.3934/cpaa.2009.8.1203 +[Abstract](204) +[PDF](821.7KB)
Abstract:
In this paper, linear degenerate parabolic diffusion equations of second order with discontinuous coefficients are studied with respect to existence and uniqueness of weak solutions. We consider the full degenerate case where the diffusion is given by a tensor field which is only positive semi-definite and essentially bounded in the whole domain. Existence of solutions in Hilbert spaces incorporating the diffusion tensor is proven and uniqueness in a certain sense is established. Moreover, we examine replacements for the missing compactness by the Lions-Aubin lemma, proving that the set of solutions associated with bounded data and bounded semi-definite coefficients is weakly relatively compact in a space of weakly continuous functions. Finally, an application to the image-processing problem of edge-preserving denoising is presented. A method based on the considered equations is introduced and numerical examples are given.
Properties and applications of a function involving exponential functions
Bai-Ni Guo and Feng Qi
2009, 8(4): 1231-1249 doi: 10.3934/cpaa.2009.8.1231 +[Abstract](595) +[PDF](246.9KB)
Abstract:
In the present paper, we give necessary and sufficient conditions for the elementary function $ q_{\alpha,\beta}(t)=\frac{e^{-\alpha t}-e^{-\beta t}}{1-e^{-t}},$ if $t\ne 0$ or $q_{\alpha,\beta}(t)=\beta-\alpha,$ if $t=0$ to be monotonic or logarithmically convex on $(-\infty,\infty)$, $(-\infty,0)$ or $(0,\infty)$ respectively, where $\alpha$ and $\beta$ are real numbers and satisfy $\alpha\ne\beta$ and $(\alpha,\beta)$ ∉ {$(0,1),(1,0)$}. Utilizing the monotonicity of $q_{\alpha,\beta}(t)$ on $(0,\infty)$, we derive necessary and sufficient conditions for the function $H_{a,b;c}(x)=(x+c)^{b-a} \frac{\Gamma(x+a)}{\Gamma(x+b)}$, its $q$-analogue, and ratios of the gamma or $q$-gamma functions to be logarithmically completely monotonic, where $a,b,c$ are real numbers and $x\in (-\min$ {$a,b,c$},$\infty)$.
Asymptotic behavior of solutions to elliptic equations in a coated body
Jingyu Li
2009, 8(4): 1251-1267 doi: 10.3934/cpaa.2009.8.1251 +[Abstract](247) +[PDF](216.8KB)
Abstract:
We consider the Dirichlet boundary-value problem for a class of elliptic equations in a domain surrounded by a thin coating with the thickness $\delta$ and the thermal conductivity $\sigma$. By virtue of a new method we further investigate the results of Brezis, Caffarelli and Friedman [3] in three respects. If the integral of the source term on the interior domain is zero, we study the asymptotic behavior of the solution in the case of $\delta^2$»$\sigma$, $\delta^2$~$\sigma$ and $\delta^2$«$\sigma$ as $\delta$ and $\sigma$ tend to zero, respectively. Also we derive the optimal blow-up rate that was not given in [3]. Finally, in the case of the so-called "optimally aligned coating", i.e., if the thermal tensor matrix of the coating is spatially varying and its smallest eigenvalue has an eigenvector normal to the body at all boundary points, we obtain the asymptotic behavior of the solution by assuming only the smallest eigenvalue is of the same order as $\sigma$.
Multiple solutions for a critical nonhomogeneous elliptic problem in domains with small holes
Salomón Alarcón
2009, 8(4): 1269-1289 doi: 10.3934/cpaa.2009.8.1269 +[Abstract](217) +[PDF](285.0KB)
Abstract:
We consider the problem: $-\Delta u=|u|^{\frac{4}{N-2}}u+\varepsilon f(x)$ in $\Omega$, $u=0$ on $\partial\Omega$, where $\Omega\subset R^N$ is a bounded smooth domain which exhibits small holes, $f\geq 0$, $f$ is not equivalent to $0$ and $\varepsilon>0$ is small. Using the reduction method and a min-max scheme worked out with topological arguments, we construct multiple solutions by gluing negative double-spike patterns located near each of the holes.
Systems of convex Hamilton-Jacobi equations with implicit obstacles and the obstacle problem
Fabio Camilli, Paola Loreti and Naoki Yamada
2009, 8(4): 1291-1302 doi: 10.3934/cpaa.2009.8.1291 +[Abstract](268) +[PDF](190.5KB)
Abstract:
Aim of this paper is to show that some of the results in the weak KAM theory for $1^{s t}$ order convex Hamilton-Jacobi equations (see [11], [13]) can be extended to systems of convex Hamilton-Jacobi equations with implicit obstacles and to the obstacle problem. We obtain two results: a comparison theorem for systems lacking strict monotonicity; a representation formula for the obstacle problem involving the distance function associated to the Hamiltonian of the equation.
Blowup and global existence of the nonlinear Schrödinger equations with multiple potentials
Zaihui Gan, Boling Guo and Jian Zhang
2009, 8(4): 1303-1312 doi: 10.3934/cpaa.2009.8.1303 +[Abstract](295) +[PDF](143.8KB)
Abstract:
This paper is concerned with the Cauchy problem for the nonlinear Schrödinger equations with multiple potentials. Under some assumptions on these potentials, we first obtain some sufficient conditions of blowup according to the initial energy and the Cauchy initial data directly. We next establish a sufficient condition of global existence by using potential well method and the relation between the Cauchy data and the ground state. We finally answer the question that how small the Cauchy initial data need to be for global existence by using scaling argument.
Hyperbolic--parabolic singular perturbation for mildly degenerate Kirchhoff equations: Global-in-time error estimates
Marina Ghisi and Massimo Gobbino
2009, 8(4): 1313-1332 doi: 10.3934/cpaa.2009.8.1313 +[Abstract](237) +[PDF](277.8KB)
Abstract:
We consider the second order Cauchy problem

$\varepsilon u_\varepsilon''+ u_\varepsilon'+m(|A^{1/2}u_\varepsilon|^2)Au_\varepsilon=0, \quad u_\varepsilon(0)=u_0,\quad u_\varepsilon'(0)=u_1,$

and the first order limit problem

$u'+m(|A^{1/2}u_\varepsilon|^2)Au=0, \quad u(0)=u_0,$

where $\varepsilon>0$, $H$ is a Hilbert space, $A$ is a self-adjoint nonnegative operator on $H$ with dense domain $D(A)$, $(u_0,u_1)\in D(A^{3/2})\times D(A^{1/2})$, and $m:[0,+\infty)\to [0,+\infty)$ is a function of class $C^1$.
We prove global-in-time estimates for the difference $u_\varepsilon(t)-u(t)$ provided that $u_0$ satisfies the nondegeneracy condition $m(|A^{1/2}u_0|^2)>0$, and the function $\sigma m(\sigma^2)$ is nondecreasing in a right neighborhood of its zeroes.
The abstract results apply to parabolic and hyperbolic partial differential equations with non-local nonlinearities of Kirchhoff type.

Existence and uniqueness results for the gradient vector flow and geodesic active contours mixed model
Laurence Guillot and Maïtine Bergounioux
2009, 8(4): 1333-1349 doi: 10.3934/cpaa.2009.8.1333 +[Abstract](249) +[PDF](248.0KB)
Abstract:
This article deals with the so called GVF (Gradient Vector Flow) introduced by C. Xu, J.L. Prince [26, 27]. We give existence and uniqueness results for the front propagation flow for boundary extraction that was initiated by Paragios, Mellina-Gottardo and Ralmesh [22, 23]. This model combines the geodesic active contour flow and the GVF to determine the geometric flow. The motion equation is considered within a level set formulation to result an Hamilton-Jacobi equation.
Asymptotic behavior of solutions for some semilinear heat equations in $R^N$
Kazuhiro Ishige and Tatsuki Kawakami
2009, 8(4): 1351-1371 doi: 10.3934/cpaa.2009.8.1351 +[Abstract](298) +[PDF](287.7KB)
Abstract:
We consider the Cauchy problem of the semilinear heat equation,

$\partial_t u = \Delta u +f(u)$ in $R^N \times (0,\infty),$

$u (x,0) = \phi (x) \ge 0$ in $R^N,\quad\quad$

where $N \geq 1$, $f \in C^1([0,\infty))$, and $\phi \in L^1(R^N) \cap L^{\infty}(R^N)$. We study the asymptotic behavior of the solutions in the $L^q$ spaces with $q \in [1,\infty]$, by using the relative entropy methods.

Long-time behaviour of doubly nonlinear parabolic equations
A. Kh. Khanmamedov
2009, 8(4): 1373-1400 doi: 10.3934/cpaa.2009.8.1373 +[Abstract](229) +[PDF](263.5KB)
Abstract:
We consider a doubly nonlinear parabolic equation in $R^n$. Under suitable hypotheses we prove that a semigroup generated by this equation possesses a global attractor.
Global existence for a forced dispersive dissipative equation via the I-method
Bassam Kojok
2009, 8(4): 1401-1419 doi: 10.3934/cpaa.2009.8.1401 +[Abstract](212) +[PDF](254.0KB)
Abstract:
In this paper, we first investigate the Korteweg-de Vries-Burgers (KdV-Burgers) equation with low regularity external force $f$, where $f$ is a space-time function. We show that we can adapt the I-method to study the global well-posedness of our problem on a Sobolev space of negative index when a low regularity space-time external force occurs.
Three nontrivial solutions for periodic problems with the $p$-Laplacian and a $p$-superlinear nonlinearity
Leszek Gasiński and Nikolaos S. Papageorgiou
2009, 8(4): 1421-1437 doi: 10.3934/cpaa.2009.8.1421 +[Abstract](218) +[PDF](222.6KB)
Abstract:
We consider a nonlinear periodic problem driven by the scalar $p$-Laplacian and a nonlinearity that exhibits a $p$-superlinear growth near $\pm\infty$, but need not satisfy the Ambrosetti-Rabinowitz condition. Using minimax methods, truncations techniques and Morse theory, we show that the problem has at least three nontrivial solutions, two of which are of fixed sign.
Uniqueness of 2-D compressible vortex sheets
Jean-françois Coulombel and Paolo Secchi
2009, 8(4): 1439-1450 doi: 10.3934/cpaa.2009.8.1439 +[Abstract](216) +[PDF](192.8KB)
Abstract:
We consider compressible vortex sheets for the isentropic Euler equations of gas dynamics in two space dimensions. Under a supersonic condition that precludes violent instabilities, in previous papers [3, 4] we have studied the linearized stability and proved the local existence of piecewise smooth solutions to the nonlinear problem. This is a free boundary nonlinear hyperbolic problem with two main difficulties: the free boundary is characteristic, and the so-called Lopatinskii condition holds only in a weak sense, which yields losses of derivatives. In the present paper we prove that sufficiently smooth solutions are unique.
Large BV solutions to Euler equations in the isothermal self-gravitating gases with damping
Yulan Xu and Yanping Dou
2009, 8(4): 1451-1467 doi: 10.3934/cpaa.2009.8.1451 +[Abstract](232) +[PDF](201.2KB)
Abstract:
The present paper gives the global existence of $BV$ solutions to the radial motions of self-gravitating gases with damping. We construct approximate solutions by a modified Glimm scheme, considering the effects of the geometric source term, the integrand term, and the damping term at the same time. With the strength of the waves measured by $|\Delta(w-z)|$, where $(w,z)$ are the Riemann invariants to the corresponding Euler equations, we prove that the $BV$ norms of the approximate solutions are bounded.

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