# American Institute of Mathematical Sciences

ISSN:
1534-0392

eISSN:
1553-5258

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## Communications on Pure & Applied Analysis

March 2009 , Volume 8 , Issue 2

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2009, 8(2): 493-508 doi: 10.3934/cpaa.2009.8.493 +[Abstract](1530) +[PDF](247.5KB)
Abstract:
We study the long-time behavior of non-negative solutions to the Cauchy problem

(P) $\qquad \rho(x) \partial_t u= \Delta u^m\qquad$ in $Q$:$=\mathbb R^n\times\mathbb R_+$

$u(x, 0)=u_0$

in dimensions $n\ge 3$. We assume that $m> 1$ (slow diffusion) and $\rho(x)$ is positive, bounded and behaves like $\rho(x)$~$|x|^{-\gamma}$ as $|x|\to\infty$, with $0\le \gamma<2$. The data $u_0$ are assumed to be nonnegative and such that $\int \rho(x)u_0 dx< \infty$.
Our asymptotic analysis leads to the associated singular equation $|x|^{-\gamma}u_t= \Delta u^m,$ which admits a one-parameter family of selfsimilar solutions $U_E(x,t)=t^{-\alpha}F_E(xt^{-\beta})$, $E>0$, which are source-type in the sense that $|x|^{-\gamma}u(x,0)=E\delta(x)$. We show that these solutions provide the first term in the asymptotic expansion of generic solutions to problem (P) for large times, both in the weighted $L^1$ sense

$u(t)=U_E(t)+o(1)\qquad$ in $L^1_\rho$

and in the uniform sense $u(t)=U_E(t)+o(t^{-\alpha})$ in $L^\infty$ as $t\to \infty$ for the explicit rate $\alpha=\alpha(m,n,\gamma)>0$ which is precisely the time-decay rate of $U_E$. For a given solution, the proper choice of the parameter is $E=\int \rho(x)u_0 dx$.

2009, 8(2): 509-531 doi: 10.3934/cpaa.2009.8.509 +[Abstract](1762) +[PDF](1485.7KB)
Abstract:
A nonlinear, density-dependent system of diffusion-reaction equations describing development of bacterial biofilms is analyzed. It comprises two non-standard diffusion effects, degeneracy as in the porous medium equation and fast diffusion. The existence of a unique bounded solution and a global attractor is proved in dependence of the boundary conditions. This is achieved by studying a system of non-degenerate auxiliary approximation equations and the construction of a Lipschitz continuous semigroup by passing to the limit in the approximation parameter. Numerical examples are included in order to illustrate the main result.
2009, 8(2): 533-557 doi: 10.3934/cpaa.2009.8.533 +[Abstract](1556) +[PDF](314.4KB)
Abstract:
We study elliptic problems at critical growth under Steklov boundary conditions in bounded domains. For a second order problem we prove existence of nontrivial nodal solutions. These are obtained by combining a suitable linking argument with fine estimates on the concentration of Sobolev minimizers on the boundary. When the domain is the unit ball, we obtain a multiplicity result by taking advantage of the explicit form of the Steklov eigenfunctions. We also partially extend the results in the ball to the case of fourth order Steklov boundary value problems.
2009, 8(2): 559-585 doi: 10.3934/cpaa.2009.8.559 +[Abstract](1932) +[PDF](367.1KB)
Abstract:
In this survey, our aim is to emphasize the main known limitations to the use of Wigner measures for Schrödinger equations. After a short review of successful applications of Wigner measures to study the semi-classical limit of solutions to Schrödinger equations, we list some examples where Wigner measures cannot be a good tool to describe high frequency limits. Typically, the Wigner measures may not capture effects which are not negligible at the pointwise level, or the propagation of Wigner measures may be an ill-posed problem. In the latter situation, two families of functions may have the same Wigner measures at some initial time, but different Wigner measures for a larger time. In the case of systems, this difficulty can partially be avoided by considering more refined Wigner measures such as two-scale Wigner measures; however, we give examples of situations where this quadratic approach fails.
2009, 8(2): 587-600 doi: 10.3934/cpaa.2009.8.587 +[Abstract](1652) +[PDF](183.5KB)
Abstract:
This paper deals with the bounded and blowup solutions of the quasilinear parabolic system $u_t = u^p ( \Delta u + a v) + f(u, v, Du, x)$ and $v_t = v^q ( \Delta v + b u) + g(u, v, Dv, x)$ with homogeneous Dirichlet boundary condition. Under suitable conditions on the lower order terms $f$ and $g$, it is shown that all solutions are bounded if $(1+c_1) \sqrt{ab} < \l_1$ and blow up in a finite time if $(1+c_1) \sqrt{ab} > \lambda_1$, where $\lambda_1$ is the first eigenvalue of $-\Delta$ in $\Omega$ with Dirichlet data and $c_1 > -1$ related to $f$ and $g$.
2009, 8(2): 601-620 doi: 10.3934/cpaa.2009.8.601 +[Abstract](1752) +[PDF](270.4KB)
Abstract:
We study an abstract inverse problem of reconstruction of the solution of a semilinear mixed integrodifferential parabolic problem, together with a convolution kernel. The supplementary information required to solve the problem also involves a convolution term with the same unknown kernel. The abstract results are applicable to the identification of a memory kernel in a strongly damped wave equation using a flux condition.
2009, 8(2): 621-644 doi: 10.3934/cpaa.2009.8.621 +[Abstract](2082) +[PDF](274.6KB)
Abstract:
In this paper the mountain--pass theorem and the Ekeland variational principle in a suitable Orlicz space are employed to establish the existence of positive standing wave solutions for a quasilinear Schrödinger equation involving a combination of concave and convex terms. The second order nonlinearity considered in this paper corresponds to the superfluid equation in plasma physics.
2009, 8(2): 645-654 doi: 10.3934/cpaa.2009.8.645 +[Abstract](1880) +[PDF](167.7KB)
Abstract:
This paper deals with the rigorous study of the diffusive stress relaxation in the multidimensional system arising in the mathematical modeling of viscoelastic materials. The control of an appropriate high order energy shall lead to the proof of that limit in Sobolev space. It is shown also as the same result can be obtained in terms of relative modulate energies.
2009, 8(2): 655-671 doi: 10.3934/cpaa.2009.8.655 +[Abstract](2075) +[PDF](215.2KB)
Abstract:
We study the stability of the dynamics of a network of $n$ formal neurons interacting through an asymmetric matrix with independent random Gaussian elements of the type introduced by Rajan and Abbott ([1]). The neurons are represented by the values of their electric potentials $x_i, i=1,\cdots,n$. Using the approach developed in a previous paper by us ([6]) we obtain sufficient conditions for diverging synchronized behavior and for stability.
2009, 8(2): 673-682 doi: 10.3934/cpaa.2009.8.673 +[Abstract](1567) +[PDF](185.3KB)
Abstract:
Let $B$ denote the unit ball in $\mathbb R^N$, $N\geq 3$. We consider the classical Brezis-Nirenberg problem

$\Delta u+\lambda u+u^{\frac{N+2}{N-2}} =0 \quad$ in$\quad B$

$u>0 \quad$ in $\quad B$

$u=0 \quad$ on $\quad \partial B$

where $\lambda$ is a constant. It is proven in [3] that this problem has a classical solution if and only if $\underline \lambda < \lambda < \lambda _1$ where $\underline \lambda = 0$ if $N\ge 4$, $\underline \lambda = \frac{\lambda _1}4$ if $N=3$. This solution is found to be unique in [17]. We prove that there is a number $\lambda_*$ and a continuous function $a(\lambda)\ge 0$ decreasing in $(\underline \lambda, \lambda_*]$, increasing in $[\lambda_*, \lambda_1)$ such that for each $\lambda$ in this range and each $\mu\in (a(\lambda),\infty)$ there exist a $\mu$-periodic function $w_\mu(t)$ and two distinct radial solutions $u_{\mu j}$, $j=1,2$, singular at the origin, with $u_{\mu j}(x)$~$|x|^{-\frac{N-2}2}w_\mu$( log $|x|$) as $x\to 0$. They approach respectively zero and the classical solution as $\mu\to +\infty$. At $\lambda =\lambda_*$ there is in addition to those above a solution ~$c_N|x|^{-\frac{N-2}2}$. This clarifies a previous result by Benguria, Dolbeault and Esteban in [2], where a existence of a continuum of singular solutions for each $\lambda\in (\underline\lambda, \lambda_1)$ was found.

2009, 8(2): 683-688 doi: 10.3934/cpaa.2009.8.683 +[Abstract](1919) +[PDF](120.3KB)
Abstract:
We provide a general framework of inequalities induced by the Aubry-Mather theory of Hamilton-Jacobi equations. This framework deals with a sufficient condition on functions $f\in C^1(\mathbb R^n)$ and $g\in C(\mathbb R^n)$ in order that $f-g$ takes its minimum over $\mathbb R^n$ on the set {$x\in \mathbb R^n |Df(x)=0$}. As an application of this framework, we provide proofs of the arithmetic mean-geometric mean inequality, Hölder's inequality and Hilbert's inequality in a unified way.
2009, 8(2): 689-710 doi: 10.3934/cpaa.2009.8.689 +[Abstract](1897) +[PDF](322.7KB)
Abstract:
We consider a phase-field system of Caginalp type on a three-dimensional bounded domain. The order parameter $\psi$ fulfills a dynamic boundary condition, while the (relative) temperature $\theta$ is subject to a boundary condition of Dirichlet, Neumann, Robin or Wentzell type. The corresponding class of initial and boundary value problems has already been studied by the authors, proving well-posedness results and the existence of global as well as exponential attractors. Here we intend to show first that the previous analysis can be redone for larger phase-spaces, provided that the bulk potential has a fourth-order growth at most whereas the boundary potential has an arbitrary polynomial growth. Moreover, assuming the potentials to be real analytic, we demonstrate that each trajectory converges to a single equilibrium by means of a Łojasiewicz-Simon type inequality. We also obtain a convergence rate estimate.
2009, 8(2): 711-718 doi: 10.3934/cpaa.2009.8.711 +[Abstract](1863) +[PDF](146.0KB)
Abstract:
In this paper, we study the blowup rate estimate for a system of semilinear parabolic equations. The blowup rate depends on whether the two components of the solution of this system blow up simultaneously or not.
2009, 8(2): 719-724 doi: 10.3934/cpaa.2009.8.719 +[Abstract](1501) +[PDF](120.6KB)
Abstract:
This work is devoted to the existence of positive entire solutions for semilinear elliptic systems. With the aid of a degree theory argument, we use the shooting method and Pohozaev-type identity to show the existence of positive radial solutions.
2009, 8(2): 725-742 doi: 10.3934/cpaa.2009.8.725 +[Abstract](1512) +[PDF](226.2KB)
Abstract:
We present two new families of polynomial differential systems of arbitrary degree with centers, a two--parameter family and a four--parameter family.
2009, 8(2): 743-755 doi: 10.3934/cpaa.2009.8.743 +[Abstract](1481) +[PDF](208.6KB)
Abstract:
We study the Schrödinger operator $H = - \Delta + V$ on the product of two copies of an infinite blowup of the Sierpinski gasket, where $V$ is the analog of a Coulomb potential ($\Delta V$ is a multiple of a delta function). So $H$ is the analog of the standard Hydrogen atom model in nonrelativistic quantum mechanics. Like the classical model, we show that the essential spectrum of $H$ is the same as for $- \Delta$, and there is a countable discrete spectrum of negative eigenvalues.
2009, 8(2): 757-768 doi: 10.3934/cpaa.2009.8.757 +[Abstract](1275) +[PDF](185.7KB)
Abstract:
For inhomogeneous diagonal system with distinct characteristics or with characteristics with constant multiplicity, under the assumption that the system is linearly degenerate and the $C^1$ norm of the initial data is bounded, we show that the mechanism of the formation of singularities of classical solution to its Cauchy problem must be of ODE type. Similar results are also obtained for corresponding mixed initial-boundary value problems on a semi-unbounded domain.
2009, 8(2): 769-783 doi: 10.3934/cpaa.2009.8.769 +[Abstract](1483) +[PDF](227.6KB)
Abstract:
In this article we prove some sharp regularity results for the stationary and the evolution Navier-Stokes equations with shear dependent viscosity, under the no-slip boundary condition. This is a classical turbulence model, considered by von Neumann and Richtmeyer in the 50's, and by Smagorinski in the beginning of the 60's (for $p=3$). The model was extended to other physical situations, and deeply studied from a mathematical point of view, by Ladyzhenskaya in the second half of the 60's. In the sequel we consider the case $p> 2$. We are interested in regularity results in Sobolev spaces, up to the boundary, in dimension $n= 3$, for the second order derivatives of the velocity and the first order derivatives of the pressure. In spite of the very rich literature on this subject, sharp regularity results up to the boundary are quite new. In the sequel we improve in a very substantial way all the known results in the literature. In order to emphasize the very new ideas, we consider a flat boundary (the so called "cubic-domain" case). However, all the regularity results stated here hold in the presence of smooth boundaries, by following [3].

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