ISSN:

1534-0392

eISSN:

1553-5258

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## Communications on Pure & Applied Analysis

January 2014 , Volume 13 , Issue 1

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2014, 13(1): 1-73
doi: 10.3934/cpaa.2014.13.1

*+*[Abstract](1227)*+*[PDF](9314.7KB)**Abstract:**

This paper analyzes the existence and structure of the positive solutions of a very simple superlinear indefinite semilinear elliptic prototype model under non-homogeneous boundary conditions, measured by $M\leq \infty$. Rather strikingly, there are ranges of values of the parameters involved in its setting for which the model admits an arbitrarily large number of positive solutions, as a result of their fast oscillatory behavior, for sufficiently large $M$. Further, using the amplitude of the superlinear term as the main bifurcation parameter, we can ascertain the global bifurcation diagram of the positive solutions. This seems to be the first work where these multiplicity results have been documented.

2014, 13(1): 75-95
doi: 10.3934/cpaa.2014.13.75

*+*[Abstract](1077)*+*[PDF](439.1KB)**Abstract:**

In this paper, some existence theorems are obtained for periodic solutions of second order Hamiltonian systems under non-quadratic conditions by using the minimax principle. Our results unite, extend and improve those relative works in some known literature.

2014, 13(1): 97-118
doi: 10.3934/cpaa.2014.13.97

*+*[Abstract](1151)*+*[PDF](452.1KB)**Abstract:**

We study the asymptotic behavior of the ground state for a class of quasilinear Schrödinger equations with general nonlinearities. By the variational argument and dual approach, we show the asymptotic non-degeneracy and uniqueness of the ground state.

2014, 13(1): 119-133
doi: 10.3934/cpaa.2014.13.119

*+*[Abstract](1237)*+*[PDF](413.7KB)**Abstract:**

We prove a comparison principle for viscosity solutions of a fully nonlinear equation satisfying a condition of non-degeneracy in a fixed direction. We apply these results to prove that a continuous solution of the corresponding Dirichlet problem exists. To obtain the existence of barrier functions and well-posedness, we find suitable explicit assumptions on the domain and on the ellipticity constants of the operator.

2014, 13(1): 135-155
doi: 10.3934/cpaa.2014.13.135

*+*[Abstract](901)*+*[PDF](435.7KB)**Abstract:**

In this paper we consider the magnetohydrodynamics flows giving rise to a variety of mathematical problems in many areas. We study the incompressible limit problems for magnetohydrodynamics flows under strong stratification on unbounded domains.

2014, 13(1): 157-173
doi: 10.3934/cpaa.2014.13.157

*+*[Abstract](1154)*+*[PDF](393.5KB)**Abstract:**

The existence and uniqueness of positive solutions of a nonautonomous system of SIR equations with diffusion are established as well as the continuous dependence of such solutions on initial data. The proofs are facilitated by the fact that the nonlinear coefficients satisfy a global Lipschitz property due to their special structure. An explicit disease-free nonautonomous equilibrium solution is determined and its stability investigated. Uniform weak disease persistence is also shown. The main aim of the paper is to establish the existence of a nonautonomous pullback attractor is established for the nonautonomous process generated by the equations on the positive cone of an appropriate function space. For this an energy method is used to determine a pullback absorbing set and then the flattening property is verified, thus giving the required asymptotic compactness of the process.

2014, 13(1): 175-202
doi: 10.3934/cpaa.2014.13.175

*+*[Abstract](1099)*+*[PDF](603.9KB)**Abstract:**

The aim of this paper is twofold. On the one hand we construct Neumann-transmission kernels for pseudodifferential Brinkman operators. They are used to provide simple representations of the solution to some transmission problems for the pseudodifferential Brinkman operator. On the other hand, we show the well-posedness of a Neumann-transmission problem for two pseudodifferential Brinkman operators on adjacent Lipschitz domains in a compact Riemannian manifold, with boundary data in some $L^p$, Sobolev or Besov spaces. We rely on the layer potential theory in order to obtain an explicit representation of the solution to this problem. Compactness and invertibility results of associated layer potential operators on $L^p$, Sobolev and Besov spaces are also presented.

2014, 13(1): 203-215
doi: 10.3934/cpaa.2014.13.203

*+*[Abstract](1113)*+*[PDF](403.4KB)**Abstract:**

We consider a parametric nonlinear Dirichlet problem driven by the $(p,q)$-differential operator, with a reaction consisting of a ``concave'' term perturbed by a $(p-1)$-superlinear perturbation, which need not satisfy the Ambrosetti-Rabinowitz condition (problem with combined or competing nonlinearities). Using variational methods we show that for small values of the parameter the problem has at least two nontrivial positive smooth solutions.

2014, 13(1): 217-223
doi: 10.3934/cpaa.2014.13.217

*+*[Abstract](875)*+*[PDF](340.8KB)**Abstract:**

A superconductive model characterized by a third order parabolic operator $ {\mathcal L}_\varepsilon $ is analyzed. When the viscous terms, represented by higher-order derivatives, tend to zero, a hyperbolic operator $ {\mathcal L}_0 $ appears. Furthermore, if ${\mathcal P}_\varepsilon$ is the Dirichlet initial-boundary value problem for $ {\mathcal L}_\varepsilon$, when ${\mathcal L} _\varepsilon $ turns into ${\mathcal L}_0 , $ ${\mathcal P}_\varepsilon$ turns into a problem ${\mathcal P}_0$ with the same initial-boundary conditions of ${\mathcal P}_\varepsilon $. As long as the higher-order derivatives of the solution of ${\mathcal P}_0$ are bounded, an estimate of solution for the nonlinear problem related to the remainder term $ r, $ is achieved. Moreover, some classes of explicit solutions related to $ {\mathcal P}_0 $ are determined, proving the existence of at least one motion whose derivatives are bounded. The estimate shows that the diffusion effects are bounded even when time tends to infinity.

2014, 13(1): 225-236
doi: 10.3934/cpaa.2014.13.225

*+*[Abstract](1177)*+*[PDF](380.5KB)**Abstract:**

We consider the existence of strong solution to liquid crystals system in critical Besov space, and give a criterion which is similar to Serrin's criterion on regularity of weak solution to Navier-Stokes equations.

2014, 13(1): 237-248
doi: 10.3934/cpaa.2014.13.237

*+*[Abstract](1151)*+*[PDF](408.4KB)**Abstract:**

In this paper, we consider the following semilinear Schrödinger equations with ciritical growth \begin{eqnarray} -\Delta u+(\lambda a(x)-\delta)u=|u|^{2^*-2}u,x\in R^N, \end{eqnarray} where $N\geq 4$, $a(x)\geq 0$ and its zero sets are not empty. $2^*$ is the critical Sobolev exponent. $\delta>0$ is a constant such that the operator $-\Delta +\lambda a(x)-\delta$ might be indefinite but is non-degenerate. We prove the existence of least energy solutions which localize near the potential well $int \{a^{-1}(0)\}$ for $\lambda$ large enough.

2014, 13(1): 249-272
doi: 10.3934/cpaa.2014.13.249

*+*[Abstract](1224)*+*[PDF](492.2KB)**Abstract:**

In this paper, we study the homogenization and corrector results for the hyperbolic problem in a two-component composite with $\varepsilon$-periodic connected inclusions. The condition prescribed on the interface is that a jump of the solution is proportional to the conormal derivatives via a function of order $\varepsilon^\gamma$ ($\gamma < -1$). The main ingredient of the proof of our main theorems is the time-dependent periodic unfolding method in two-component domains. Our homogenization results recover those of the corresponding case in [Donato, Faella and Monsurrò, J. Math. Pures Appl. 87 (2007), pp. 119-143]. We also derive the corresponding corrector results.

2014, 13(1): 273-291
doi: 10.3934/cpaa.2014.13.273

*+*[Abstract](1512)*+*[PDF](515.0KB)**Abstract:**

Extending previous works [47, 27, 30], we consider in even space dimensions the initial value problems for some high-order semi-linear wave and Schrödinger type equations with exponential nonlinearity. We obtain global well-posedness in the energy space.

2014, 13(1): 293-305
doi: 10.3934/cpaa.2014.13.293

*+*[Abstract](1687)*+*[PDF](401.2KB)**Abstract:**

This paper deals with the weighted nonlinear elliptic equation \begin{eqnarray} -\mathrm{div}(|x|^\alpha \nabla u )=|x|^\gamma e^u \ in\ \Omega ,\\ u = 0 \ on \ \partial \Omega, \end{eqnarray} where $\alpha, \gamma \in R$ satisfy $N + \alpha > 2$ and $\gamma - \alpha > -2$, and the domain $\Omega \subset R^N (N \geq 2)$ is bounded or not. Moreover, when $\alpha\neq 0$, we prove that, for $N + \alpha > 2$, $\gamma - \alpha \leq -2$, the above equation admits no weak solution. We also study Liouville type results for the equation in $R^N$.

2014, 13(1): 307-330
doi: 10.3934/cpaa.2014.13.307

*+*[Abstract](1136)*+*[PDF](470.4KB)**Abstract:**

In this paper, we study the time-asymptotic behavior of the solution for the Cauchy problem of the damped wave equation with a nonlinear convection term in the multi-dimensional space. When the initial data is a small perturbation around a constant state $u^*$, we obtain the point-wise decay estimates of the solution under the so-called dissipative condition $|b| < 1$, where $b$ depends on $u^*$ and the nonlinear term.

2014, 13(1): 331-346
doi: 10.3934/cpaa.2014.13.331

*+*[Abstract](1096)*+*[PDF](385.9KB)**Abstract:**

Extending previuos results ([16, 1, 7]), we study the vanishing viscosity limit of solutions of space-time periodic Hamilton-Jacobi-Belllman equations, assuming that the ``Aubry set'' is the union of a finite number of hyperbolic periodic orbits of the Hamiltonian flow.

2014, 13(1): 347-369
doi: 10.3934/cpaa.2014.13.347

*+*[Abstract](1499)*+*[PDF](468.0KB)**Abstract:**

In this paper, a homogeneous reaction-diffusion model describing the control growth of mammalian hair is investigated. We provide some global analyses of the model depending upon some parametric thresholds/constraints. We find that when one of the dimensionless parameter is less than one, then the unique positive equilibrium is globally asymptotically stable. On the contrary, when this threshold is greater than one, the existence of both steady-state and Hopf bifurcations can be observed under further parametric constraints. In addition, we find that both spatially homogeneous and heterogeneous oscillatory solutions can be seen for some spatially independent parameters provided that some conditions are met. Under these conditions, the direction and stability of these oscillatory behaviors, global stability of the unique constant steady state and the local orbital asymptotic stability of the spatially homogeneous periodic orbits are also investigated.

2014, 13(1): 371-387
doi: 10.3934/cpaa.2014.13.371

*+*[Abstract](969)*+*[PDF](395.2KB)**Abstract:**

We consider a mathematical model that describes frictionless contact between a viscoplastic body and a deformable obstacle or foundation. The process is quasistatic and contact is modeled with the

*normal compliance with limited penetration*condition, which has been introduced recently. Moreover, the contact stiffness coefficient is allowed to depend on the history of the contact process. We derive a variational formulation of the problem, which is in the form of a strongly nonlinear system coupling an integral equation and a time-dependent variational inequality. Then, we provide the analysis of the problem, which includes its unique weak solvability and the continuous dependence of the solution on the problem data. The proofs are based on results from the theory of history-dependent variational inequalities, on monotonicity and a fixed point argument.

2014, 13(1): 389-418
doi: 10.3934/cpaa.2014.13.389

*+*[Abstract](1092)*+*[PDF](537.9KB)**Abstract:**

We prove polynomial-in-time upper bounds for the orbital instability of solitons for subcritical generalized Korteweg-de Vries equations in $H_x^s R$ with $s < 1$. By combining coercivity estimates of Weinstein with the $I$-method as developed by Colliander, Keel, Staffilani, Takaoka, and Tao, we construct a modified energy functional which is shown to be almost conserved while providing us with an estimate of the deviation of the solution from the ground state curve. The iteration of the almost conservation law for the modified energy functional over time intervals of uniform length yields the polynomial upper bound.

2014, 13(1): 419-433
doi: 10.3934/cpaa.2014.13.419

*+*[Abstract](1068)*+*[PDF](409.0KB)**Abstract:**

Let $\Omega$ be a smooth bounded domain in $R^N$ and let \begin{eqnarray} Lu=\sum_{j,k=1}^N \partial_{x_j}\left(a_{jk}(x)\partial_{x_k} u\right), \end{eqnarray} in $\Omega$ and \begin{eqnarray} Lu+\beta(x)\sum\limits_{j,k=1}^N a_{jk}(x)\partial_{x_j} u n_k+\gamma (x)u-q\beta(x)\sum_{j,k=1}^{N-1}\partial_{\tau_k}\left(b_{jk}(x)\partial_{\tau_j}u\right)=0, \end{eqnarray} on $\partial\Omega$ define a generalized Laplacian on $\Omega$ with a Wentzell boundary condition involving a generalized Laplace-Beltrami operator on the boundary. Under some smoothness and positivity conditions on the coefficients, this defines a nonpositive selfadjoint operator, $-S^2$, on a suitable Hilbert space. If we have a sequence of such operators $S_0,S_1,S_2,...$ with corresponding coefficients \begin{eqnarray} \Phi_n=(a_{jk}^{(n)},b_{jk}^{(n)}, \beta_n,\gamma_n,q_n) \end{eqnarray} satisfying $\Phi_n\to\Phi_0$ uniformly as $n\to\infty$, then $u_n(t)\to u_0(t)$ where $u_n$ satisfies \begin{eqnarray} i\frac{du_n}{dt}=S_n^m u_n, \end{eqnarray} or \begin{eqnarray} \frac{d^2u_n}{dt^2}+S_n^{2m} u_n=0, \end{eqnarray} or \begin{eqnarray} \frac{d^2u_n}{dt^2}+F(S_n)\frac{du_n}{dt}+S_n^{2m} u_n=0, \end{eqnarray} for $m=1,2,$ initial conditions independent of $n$, and for certain nonnegative functions $F$. This includes Schrödinger equations, damped and undamped wave equations, and telegraph equations.

2014, 13(1): 435-452
doi: 10.3934/cpaa.2014.13.435

*+*[Abstract](888)*+*[PDF](522.7KB)**Abstract:**

We give a complete answer to the question of when two curves in two different Riemannian manifolds can be seen as trajectories of rolling one manifold on the other without twisting or slipping. We show that, up to technical hypotheses, a rolling along these curves exists if and only if the geodesic curvatures of each curve coincide. By using the anti-developments of the curves, which we claim can be seen as a generalization of the geodesic curvatures, we are able to extend the result to arbitrary absolutely continuous curves. For a manifold of constant sectional curvature rolling on itself, two such curves can only differ by an isometry. In the case of surfaces, we give conditions for when loops in the manifolds lift to loops in the configuration space of the rolling.

2014, 13(1): 453-481
doi: 10.3934/cpaa.2014.13.453

*+*[Abstract](1367)*+*[PDF](1288.9KB)**Abstract:**

We compute the Lie symmetry algebra of the equation of Helfrich surfaces and we show that it is the algebra of conformal vector fields of $R^2$. We also show that in the particular case of the Willmore surfaces we have to add the homothety vector field of $R^3$ to the aforementioned algebra. We prove that a Helfrich surface that is invariant w.r.t. a conformal symmetry is a helicoid and that all such surface solutions satisfy one and the same system of ordinary differential equations obtained by symmetry reduction. We also show that for the Willmore surface shape equation the symmetry reduction leads to two systems of ODEs. Then we construct explicit solutions in the case of revolution surfaces. The results obtained can be extended to the study of PDE problems in $2$ spatial dimensions admitting conformal Lie symmetries.

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