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Communications on Pure & Applied Analysis

2005 , Volume 4 , Issue 1

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Regularity of solutions for a system of integral equations
Wenxiong Chen and Congming Li
2005, 4(1): 1-8 doi: 10.3934/cpaa.2005.4.1 +[Abstract](257) +[PDF](183.9KB)
Abstract:
In this paper, we study positive solutions of the following system of integral equations in $R^n$:

$u(x) = \int_{R^{n}} |x-y|^{\alpha -n} v(y)^q dy$, $ v(x) = \int_{R^{n}} |x-y|^{\alpha -n} u(y)^p dy$

with $\frac{1}{q+1}+\frac{1}{p+1}=\frac{n-\alpha}{n}$. In our previous paper, under the natural integrability conditions $u \in L^{p+1} (R^n)$ and $v \in L^{q+1} (R^n)$, we prove that all the solutions are radially symmetric and monotone decreasing about some point. In this paper, we go further to study the regularity of the solutions. We show that the solutions are bounded, and hence continuous and smooth. We also prove that if $p = q$, then $u = v$, and they both must assume the standard form

$ c(\frac{t}{t^2 + |x - x_o|^2})^{(n-\alpha)/2} $

with some constant $c = c(n, \alpha)$, and for some $t > 0$ and $x_o \in R^n$.

On the stationary solutions of generalized reaction diffusion equations with $p\& q$-Laplacian
L. Cherfils and Y. Il'yasov
2005, 4(1): 9-22 doi: 10.3934/cpaa.2005.4.9 +[Abstract](330) +[PDF](225.5KB)
Abstract:
In this paper, we analyse a family of stationary nonlinear equations with $p\& q$- Laplacian $-\Delta_p u -\Delta_q u=\lambda c(x,u)$ which have a wide spectrum of applications in many areas of science. We introduce a new type of variational principles corresponding to this family of equations. Using this formalism, we exhibit intervals for the scalar parameter $\lambda$ where there exist positive solutions of the considered problems. Furthermore, we prove, in another interval, the nonexistence of nontrivial solutions. These results are different from those of existence and nonexistence for stationary equations with single Laplacian.
Existence of steady flows of slightly compressible viscoelastic fluids of White-Metzner type around an obstacle
Colette Guillopé, Abdelilah Hakim and Raafat Talhouk
2005, 4(1): 23-43 doi: 10.3934/cpaa.2005.4.23 +[Abstract](185) +[PDF](261.5KB)
Abstract:
This work is concerned with the study of steady flows, around an obstacle, of slightly compressible viscoelastic fluids, for which the extra-stress tensor is given by a White-Metzner constitutive law. The existence and uniqueness of such flows are shown, when Newtonian viscosity is present (Jeffreys' model), and for small data.
Maximal function estimates with applications to a modified Kadomstev-Petviashvili equation
C. E. Kenig and S. N. Ziesler
2005, 4(1): 45-91 doi: 10.3934/cpaa.2005.4.45 +[Abstract](310) +[PDF](363.9KB)
Abstract:
In this paper we give optimal (up to endpoint) maximal function type estimates for the solution of the linear initial value problem associated with the Kadomstev-Petviashvili I equation. These estimates enable us to obtain a well-posedness result for a modified Kadomstev-Petviashvili I equation.
Gamma-star-shapedness for semilinear elliptic equations
Antonio Greco and Marcello Lucia
2005, 4(1): 93-99 doi: 10.3934/cpaa.2005.4.93 +[Abstract](174) +[PDF](176.1KB)
Abstract:
A generic semilinear equation in a star-shaped ring is considered. Any solution bounded between its boundary values is shown to be decreasing along rays starting from the origin, provided that a structural condition is satisfied. A corresponding property for the product between the solution and a (positive) power of $|x|$ is also investigated. Applications to the Emden-Fowler and the Liouville equation are developed.
The Faber--Krahn inequality for random/nonautonomous parabolic equations
Janusz Mierczyński and Wenxian Shen
2005, 4(1): 101-114 doi: 10.3934/cpaa.2005.4.101 +[Abstract](177) +[PDF](223.0KB)
Abstract:
The paper extends the Faber--Krahn inequality for elliptic and periodic parabolic problems to random and general nonautonomous parabolic problems. Under proper assumptions, it also provides necessary and sufficient conditions for the Faber-Krahn inequality being equality.
Trajectory and global attractors of dissipative hyperbolic equations with memory
V. V. Chepyzhov and A. Miranville
2005, 4(1): 115-142 doi: 10.3934/cpaa.2005.4.115 +[Abstract](207) +[PDF](295.0KB)
Abstract:
We consider in this article a general construction of trajectory attractors and global attractors of evolution equations with memory. In our approach, the corresponding dynamical system acts in the space of initial data of the Cauchy problem under study; we can note that, in previous studies, the so-called history space setting was introduced and the study of global attractors was made in an extended phase space.
As an application, we construct trajectory and global attractors for dissipative hyperbolic equations with linear memory. We also prove the existence of a global Lyapunov function for the dissipative hyperbolic equation with memory. The existence of such a Lyapunov function implies a regular structure for the trajectory and global attractors of the equation under consideration.
Existence of sign changing solutions for some critical problems on $\mathbb R^N$
Norimichi Hirano, A. M. Micheletti and A. Pistoia
2005, 4(1): 143-164 doi: 10.3934/cpaa.2005.4.143 +[Abstract](189) +[PDF](272.9KB)
Abstract:
The main purpose of this paper is to construct families of positive and changing-sign solutions for both the slightly subcritical and slightly supercritical equations

$-\Delta u+V(x)u=N(N-2)|u|^{\frac{4}{N-2}\pm\varepsilon}u$ in $\mathbb R^N,$

which blow-up and concentrate at different points of $\mathbb R^N$ as $\varepsilon$ goes to 0, under certain conditions on the potential $V.$

On the dimension of the attractor for the wave equation with nonlinear damping
Dalibor Pražák
2005, 4(1): 165-174 doi: 10.3934/cpaa.2005.4.165 +[Abstract](252) +[PDF](204.3KB)
Abstract:
We give an explicit estimate of the fractal dimension of the global attractor to the wave equation with nolinear damping. The nonlinearities are smooth functions of certain polynomial growth. As a by-product we estimate the dimension of the exponential attractor for the time $\tau$ solution operator provided that $\tau$ is sufficiently large. The main tool used in the proof is the so-called method of the trajectories.
Solutions of minimal period for a Hamiltonian system with a changing sign potential
Yavdat Il'yasov and Nadir Sari
2005, 4(1): 175-185 doi: 10.3934/cpaa.2005.4.175 +[Abstract](169) +[PDF](209.9KB)
Abstract:
We consider a class of second-order Hamiltonian systems with a potential indefinite in sign. Applying the fibering approach we prove some existence and multiplicity results of periodic solutions with minimal period. We also give an answer to the problem of the existence of solutions with prescribed period $T$ which is greater than the first eigenvalue $\frac{2\pi}{\omega_n}$ of the corresponding linear problem.
On the existence and asymptotic behavior of large solutions for a semilinear elliptic problem in $R^n$
Haitao Yang
2005, 4(1): 187-198 doi: 10.3934/cpaa.2005.4.197 +[Abstract](346) +[PDF](199.0KB)
Abstract:
In this paper, we give an existence result for nonradial large solutions of the semilinear elliptic equation $\Delta u =p(x)f(u)$ in $R^N (N\ge 3)$, where $f$ is assumed to satisfy $(f_1)$ and $(f_2)$ below. The asymptotic behavior of the large solutions at infinity are also studied in the sublinear case that $f(u)$ behaves like $u^{\gamma}$ at $\infty$ for $\gamma \in (0, 1)$.
Uniqueness and existence of maximal and minimal solutions of fully nonlinear elliptic PDE
Robert Jensen and Andrzej Świech
2005, 4(1): 199-207 doi: 10.3934/cpaa.2005.4.187 +[Abstract](263) +[PDF](199.4KB)
Abstract:
Two results are proved in the paper. The first is a uniqueness theorem for viscosity solutions of Dirichlet boundary value problems for Bellman-Isaacs equations with just measurable lower order terms. The second is a proof that there always exist maximal and minimal viscosity solutions of Dirichlet boundary value problems for fully nonlinear, uniformly elliptic PDE that are measurable in the $x$-variable.

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