# American Institute of Mathematical Sciences

ISSN:
1534-0392

eISSN:
1553-5258

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## Communications on Pure & Applied Analysis

2008 , Volume 7 , Issue 4

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2008, 7(4): 745-763 doi: 10.3934/cpaa.2008.7.745 +[Abstract](186) +[PDF](381.2KB)
Abstract:
We study nontrivial stationary solutions to a nonlinear boundary value problem with parameter $\varepsilon>0$ and the corresponding linearized eigenvalue problem. By using a particular solution of a linear ordinary differential equation of the third order, we give expressions of all eigenvalues and eigenfunctions to the linearized problems. They are completely determined by a characteristic function which consists of complete elliptic integrals. We also show asymptotic formulas of eigenvalues with respect to sufficiently small $\varepsilon$. These results give important information for profiles of corresponding eigenfunctions.
2008, 7(4): 765-786 doi: 10.3934/cpaa.2008.7.765 +[Abstract](194) +[PDF](277.3KB)
Abstract:
We consider the following problem

$-\Delta u=\frac{\lambda}{(1-u)^2}$ in $\Omega$, $u=0$ on $\partial \Omega$, $0 < u < 1$ in $\Omega$

where $\Omega$ is a rather symmetric domain in $\mathbb R^2$. We prove that there exists a $\lambda_\star>0$ such that for $\lambda \in (0, \lambda_\star)$ the minimal solution is unique. Then we analyze the asymptotic behavior of touch-down solutions, i.e., solutions with max$_\Omega u_i (0) \to 1$. We show that after a rescaling, the solution will be asymptotically symmetric. As a consequence, we show that the branch of positive solutions must undergo infinitely many bifurcations as the maximums of the solutions on the branch go to 1 (possibly only changes of direction). This gives a positive answer to some open problems in [12]. Our result is new even in the radially symmetric case. Central to our analysis is the monotonicity formula, one-dimensional Sobloev inequality, and classification of solutions to a supercritical problem

$\Delta U=\frac{1}{U^2}\quad$ in $\mathbb R^2, U(0)=1, U(z) \geq 1.$

2008, 7(4): 787-793 doi: 10.3934/cpaa.2008.7.787 +[Abstract](169) +[PDF](133.0KB)
Abstract:
We prove a conjecture by Dalbono-McKenna on the number of solutions for a weakly coupled elliptic system. The system is of the Ambrosetti-Prodi type with a asymmetric nonlinearity. We consider the radial case in a ball. By applying a degree theoretic argument, we simplify the proof of the paper [3] and obtain the existence and multiplicity of solutions for this weakly coupled system.
2008, 7(4): 795-817 doi: 10.3934/cpaa.2008.7.795 +[Abstract](188) +[PDF](295.4KB)
Abstract:
We consider

$\Delta u = 0$ in $\Omega$, $\qquad \frac{\partial u}{\partial \nu} =\lambda f(u)$ on $\Gamma_1, \qquad u = 0$ on $\Gamma_2$

where $\lambda>0$, $f(u) = e^u$ or $f(u) = (1+u)^p$, $\Gamma_1$, $\Gamma_2$ is a partition of $\partial \Omega$ and $\Omega\subset \mathbb R^N$. We determine sharp conditions on the dimension $N$ and $p>1$ such that the extremal solution is bounded, where the extremal solution refers to the one associated to the largest $\lambda$ for which a solution exists.

2008, 7(4): 819-836 doi: 10.3934/cpaa.2008.7.819 +[Abstract](189) +[PDF](278.5KB)
Abstract:
In this article, we construct a robust (that is, lower and upper semi-continuous) family of exponential attractors for a conserved Cahn-Hilliard model with the perturbation parameter in the boundary conditions. We note that the existence of a global attractor with finite dimension follows. Moreover, we prove the upper semi-continuity of the limiting attractor with respect to the family of perturbed global attractors.
2008, 7(4): 837-844 doi: 10.3934/cpaa.2008.7.837 +[Abstract](173) +[PDF](127.1KB)
Abstract:
An integral equation of contact problem of the theory of visco elasticity of mixed Fredholm and Volterra type with spectral parameter depending on time is considered. In the case where the final value of parameter coincides with some isolated point of the spectrum of Fredholm operator the additional conditions of solvability are established.
2008, 7(4): 845-852 doi: 10.3934/cpaa.2008.7.845 +[Abstract](199) +[PDF](148.1KB)
Abstract:
We consider the local existence and blowup criterion of the 3D Lagrangian averaged Euler equations in Besov spaces and obtain the existence and blowup criterion.
2008, 7(4): 853-865 doi: 10.3934/cpaa.2008.7.853 +[Abstract](155) +[PDF](186.3KB)
Abstract:
In this paper we study the limit of Monge-Kantorovich mass transfer problems when the involved measures are supported in a small strip near the boundary of a bounded smooth domain, $\Omega$. Given two absolutely continuos measures (with respect to the surface measure) supported on the boundary $\partial \Omega$, by performing a suitable extension of the measures to a strip of width $\varepsilon$ near the boundary of the domain $\Omega$ we consider the mass transfer problem for the extensions. Then we study the limit as $\varepsilon$ goes to zero of the Kantorovich potentials for the extensions and obtain that it coincides with a solution of the original mass transfer problem. Moreover we look for the possible approximations of these problems by solutions to equations involving the $p-$Laplacian for large values of $p$.
2008, 7(4): 867-881 doi: 10.3934/cpaa.2008.7.867 +[Abstract](209) +[PDF](227.9KB)
Abstract:
We study the dispersive Kuramoto-Sivashinsky and Kuramoto-Velarde equations. We show that the associated initial value problem is locally (and globally in some cases) well-posed in Sobolev spaces $H^s(\mathbb R)$ for $s > -1$. We also prove that these results are sharp in the sense that the flow map of these equations fails to be $C^2$ in $H^s(\mathbb R)$ for $s < -1$. In addition, we determine the limiting behavior of the solutions when the dispersive parameter tends to zero.
2008, 7(4): 883-903 doi: 10.3934/cpaa.2008.7.883 +[Abstract](234) +[PDF](216.7KB)
Abstract:
We study the nonlinear Schrödinger equations:

$-\epsilon^2 \Delta u+V(x)u=f(u), \quad u\in H^1(\mathbb R^N),$ $\qquad\qquad\qquad$ (*)$_\epsilon$

where $V$ satisfies $V(x)\geq 0$, liminf$_{ |x|\to \infty }V(x)>0$. $f(u)\in C^1(\mathbb R,\mathbb R)$ satisfies Ambrosetti--Rabinowitz condition and some properties and $\frac{f(u)}{u}$ is nondecreasing. We consider the case inf$_{x\in \mathbb R^N}V(x)>0$ and critical frequency case, that is, inf$_{x\in \mathbb R^N}V(x)=0$. We study the existence of sign-changing 2-peak solutions of (*)$_\epsilon$ whose one peak is positive, another peak is negative and both peaks concentrate to a same local minimum point of $V(x)$ as $\epsilon \to 0$.

2008, 7(4): 905-923 doi: 10.3934/cpaa.2008.7.905 +[Abstract](180) +[PDF](246.4KB)
Abstract:
We obtain the existence results of nodal solutions for singular one-dimensional $\varphi$-Laplacian problem with asymptotic condition:

$\varphi (u'(t))' + \lambda h(t) f (u(t)) = 0,\ \$ a.e. $\ t \in (0,1), \qquad\qquad\qquad\qquad\qquad$ $(\Phi_\lambda)$

$u(0) = 0=u(1),$

where $\varphi : \mathbb R \to \mathbb R$ is an odd increasing homeomorphism, $\lambda$ a positive parameter and $h \in L^1(0,1)$ a nonnegative measurable function on $(0,1)$ which may be singular at $t = 0$ and/or $t = 1,$ and $f \in C(\mathbb R, \mathbb R)$ and is odd.

2008, 7(4): 925-946 doi: 10.3934/cpaa.2008.7.925 +[Abstract](181) +[PDF](279.5KB)
Abstract:
We investigate here the elliptic equation -div$(a(x)\nabla u)+a(x)u=0$ posed on a bounded smooth domain $\Omega$ in $\mathbb R^2$ with nonlinear Neumann boundary condition $\frac{\partial u}{\partial \nu}=\varepsilon e^u$, where $\varepsilon$ is a small parameter. We extend the work of Davila-del Pino-Musso [5] and show that if a family of solutions $u_\varepsilon$ for which $\varepsilon\int_{\partial Omega}e^{u_\varepsilon}$ is bounded, then it will develop up to subsequences a finite number of bubbles $\xi_i\in\partial Omega$, in the sense that $\varepsilon e^{u_\varepsilon}\to 2\pi\sum_{i=1}^k m_i\delta_{\xi_i}$ as $\varepsilon\rightarrow 0$ with $k, m_i \in \mathbb Z^+$. Location of blow-up points is characterized in terms of function $a(x)$.
2008, 7(4): 947-970 doi: 10.3934/cpaa.2008.7.947 +[Abstract](260) +[PDF](293.6KB)
Abstract:
In this work the existence of a global attractor for the solution semiflow of the Gray-Scott equations with the Neumann boundary conditions on bounded domains of space dimensions $n\leq 3$ is proved. This reaction-diffusion system does not have dissipative property inherently due to the oppositely signed nonlinearity. The asymptotical compactness is shown by a new decomposition method. It is also proved that the Hausdorff dimension and the fractal dimension of the global attractor are finite.
2008, 7(4): 971-985 doi: 10.3934/cpaa.2008.7.971 +[Abstract](126) +[PDF](164.3KB)
Abstract:
In this paper we obtain sharp upper estimates on the uniform Lyapunov dimension of a cascade system in terms of the corresponding Lyapunov exponents of their components. The obtained result is applied for estimating the Lyapunov and fractal dimensions of the attractors of nonautonomous dissipative systems generated by PDEs of mathematical physics.
2008, 7(4): 987-1016 doi: 10.3934/cpaa.2008.7.987 +[Abstract](184) +[PDF](329.9KB)
Abstract:
Local solutions of the multidimensional Navier-Stokes equations for isentropic compressible flow are constructed with spherically symmetric initial data between a solid core and a free boundary connected to a surrounding vacuum state. The viscosity coefficients $\lambda, \mu$ are proportional to $\rho^\theta$, $0<\theta<\gamma$, where $\rho$ is the density and $\gamma > 1$ is the physical constant of polytropic fluid. It is also proved that no vacuum develops between the solid core and the free boundary, and the free boundary expands with finite speed.

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