ISSN:

1534-0392

eISSN:

1553-5258

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## Communications on Pure & Applied Analysis

2016 , Volume 15 , Issue 1

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2016, 15(1): 1-8
doi: 10.3934/cpaa.2016.15.1

*+*[Abstract](316)*+*[PDF](816.0KB)**Abstract:**

The difference between the number of lattice points in a disk of radius $\sqrt{t}/2\pi$ and the area of the disk $t/4\pi$ is equal to the error in the Weyl asymptotic estimate for the eigenvalue counting function of the Laplacian on the standard flat torus. We give a sharp asymptotic expression for the average value of the difference over the interval $0 \leq t \leq R$. We obtain similar results for families of ellipses. We also obtain relations to the eigenvalue counting function for the Klein bottle and projective plane.

2016, 15(1): 9-39
doi: 10.3934/cpaa.2016.15.9

*+*[Abstract](399)*+*[PDF](562.2KB)**Abstract:**

Let $N(t)$ denote the eigenvalue counting function of the Laplacian on a compact surface of constant nonnegative curvature, with or without boundary. We define a refined asymptotic formula $\widetilde N(t)=At+Bt^{1/2}+C$, where the constants are expressed in terms of the geometry of the surface and its boundary, and consider the average error $A(t)=\frac 1 t \int^t_0 D(s)\,ds$ for $D(t)=N(t)-\widetilde N(t)$. We present a conjecture for the asymptotic behavior of $A(t)$, and study some examples that support the conjecture.

2016, 15(1): 41-55
doi: 10.3934/cpaa.2016.15.41

*+*[Abstract](500)*+*[PDF](437.7KB)**Abstract:**

We study the large time behavior of small solutions to the Cauchy problem for a nonlinear damped wave equation. We proved that the solution is approximated by the Gauss kernel with suitable choice of the coefficients and powers of $t$ for $N+1$ th order for all $N \in \mathbb{N}$. Our analysis is based on the approximation theorem of the linear solution by the solution of the heat equation [37]. In particular, as pointed out by Galley-Raugel [4], we explicitly observe that from third order expansion, the asymptotic behavior of the solutions of a nonlinear damped wave equation is different from that of a nonlinear heat equation.

2016, 15(1): 57-72
doi: 10.3934/cpaa.2016.15.57

*+*[Abstract](402)*+*[PDF](450.2KB)**Abstract:**

In this paper, we study the existence of homoclinic solutions to the following second-order Hamiltonian systems \begin{eqnarray} \ddot{u}(t)-L(t)u(t)+\nabla W(t,u(t))=0,\quad \forall t\in R, \end{eqnarray} where $L(t)$ is a symmetric and positive definite matrix for all $t\in R$. The nonlinear potential $W$ is a combination of superlinear and sublinear terms. By different conditions on the superlinear and sublinear terms, we obtain existence and nonuniqueness of nontrivial homoclinic solutions to above systems.

2016, 15(1): 73-90
doi: 10.3934/cpaa.2016.15.73

*+*[Abstract](475)*+*[PDF](465.0KB)**Abstract:**

In this paper, we study the large-time behavior of weak solutions to the initial-boundary problem arising in a simplified Ericksen-Leslie system for nonhomogeneous incompressible flows of nematic liquid crystals with a transformation condition of trigonometric functions (called by trigonometric condition for simplicity) posed on the initial direction field in a bounded domain $\Omega\subset \mathbb{R}^2$. We show that the kinetic energy and direction field converge to zero and an equilibrium state, respectively, as time goes to infinity. Further, if the initial density is away from vacuum and bounded, then the density, and velocity and direction fields exponential decay to an equilibrium state. In addition, we also show that the weak solutions of the corresponding compressible flows converge {an equilibrium} state.

2016, 15(1): 91-102
doi: 10.3934/cpaa.2016.15.91

*+*[Abstract](396)*+*[PDF](406.4KB)**Abstract:**

In this paper, by computing the relevant critical groups, we obtain nontrivial solutions via Morse theory to the nonlocal $p$-Kirchhoff-type quasilinear elliptic equation \begin{eqnarray} (P)\quad\quad &&\displaystyle\bigg[M\bigg(\int_\Omega|\nabla u|^p dx\bigg)\bigg]^{p-1}(-\Delta_pu) = f(x,u), \quad x\in\Omega,\\ && u=0, \quad x\in \partial \Omega, \end{eqnarray} where $\Omega \subset \mathbb R^N$ is a bounded open domain with smooth boundary $\partial \Omega$ and $N \geq 3$.

2016, 15(1): 103-125
doi: 10.3934/cpaa.2016.15.103

*+*[Abstract](491)*+*[PDF](508.3KB)**Abstract:**

We are concerned with standing waves for the following Schrödinger-Poisson equation with critical nonlinearity: \begin{eqnarray} && - {\varepsilon ^2}\Delta u + V(x)u + \psi (x)u = \lambda W(x){\left| u \right|^{p - 2}}u + {\left| u \right|^4}u\;\;{\text{ in }}\mathbb{R}^3, \\ && - {\varepsilon ^2}\Delta \psi = {u^2}\;\;{\text{ in }}\mathbb{R}^3, u>0, u \in {H^1}(\mathbb{R}^3), \end{eqnarray} where $\varepsilon $ is a small positive parameter, $\lambda > 0$, $3 < p \le 4$, $V$ and $W$ are two potentials. Under proper assumptions, we prove that for $\varepsilon > 0$ sufficiently small, the above problem has a positive ground-state solution ${u_\varepsilon }$ by using a monotonicity trick and a new version of global compactness lemma. Moreover, we use another global compactness method due to [C. Gui, Commun. Partial Differential Equations 21 (1996) 787-820] to show that ${u_\varepsilon }$ concentrates around a set which is related to the set where the potential $V(x)$ attains its global minima or the set where the potential $W(x)$ attains its global maxima as $\varepsilon \to 0$. As far as we know, the existence and concentration behavior of the positive solutions to the Schrödinger-Poisson equation with critical nonlinearity $g(u): = \lambda W(x)|u{|^{p - 2}}u + |u{|^4}u$ $(3

2016, 15(1): 127-138
doi: 10.3934/cpaa.2016.15.127

*+*[Abstract](412)*+*[PDF](395.3KB)**Abstract:**

The Riemann problem for the relativistic generalized Chaplygin Euler equations is considered. Its two characteristic fields are genuinely nonlinear, but the nonclassical solutions appear. The formation of mechanism for $\delta-$shock is analyzed, that is the one-shock curve and the two-shock curve do not intersect each other in the phase plane. The Riemann solutions are constructed, and the generalized Rankine-Hugoniot conditions and the $\delta-$entropy condition are clarified. Moreover, under the generalized Rankine-Hugoniot conditions and entropy condition, we constructively obtain $\delta-$shock waves.

2016, 15(1): 139-160
doi: 10.3934/cpaa.2016.15.139

*+*[Abstract](444)*+*[PDF](499.7KB)**Abstract:**

This paper is to study traveling fronts of reaction-diffusion equations with space-time periodic advection and nonlinearity in $\mathbb{R}^N$ with $N\geq3$. We are interested in curved fronts satisfying some ``pyramidal" conditions at infinity. In $\Bbb{R}^3$, we first show that there is a minimal speed $c^{*}$ such that curved fronts with speed $c$ exist if and only if $c\geq c^{*}$, and then we prove that such curved fronts are decreasing in the direction of propagation. Furthermore, we give a generalization of our results in $\mathbb{R}^N$ with $N\geq4$.

2016, 15(1): 161-183
doi: 10.3934/cpaa.2016.15.161

*+*[Abstract](422)*+*[PDF](487.9KB)**Abstract:**

In this paper, we establish a Serrin-type blowup criterion for the Cauchy problem of the three dimensional compressible Navier-Stokes-Maxwell system, which states a classical solution exists globally, provided that the velocity satisfies Serrin's condition and that the $L_t^\infty L_x^\infty$ of density $\rho$ and the $L^2_tL_x^2$ of $\nabla^2 E$ are bounded. In particular, this criterion is analogous to the well-known Serrin's blowup criterion for the three-dimensional compressible Navier-Stokes equations. Moreover, it is independent of the temperature and magnetic field. It should be noted that it is the first result about the possibility of global existence of classical solution for the full Navier-Stokes-Maxwell system.

2016, 15(1): 185-196
doi: 10.3934/cpaa.2016.15.185

*+*[Abstract](416)*+*[PDF](390.1KB)**Abstract:**

Sobolev type equation theory has been an object of interest in recent years, with much attention being devoted to deterministic equations and systems. Still, there are also mathematical models containing random perturbation, such as white noise; these models are often used in natural experiments and have recently driven a large amount of research on stochastic differential equations. A new concept of ``white noise", originally constructed for finite dimensional spaces, is extended here to the case of infinite dimensional spaces. The main purpose is to develop stochastic higher-order Sobolev type equation theory and provide some practical applications. The main idea is to construct ``noise" spaces using the Nelson -- Gliklikh derivative. Abstract results are applied to the Boussinesq -- Lòve model with additive ``white noise" within Sobolev type equation theory. Because of their usefulness, we mainly focus on Sobolev type equations with relatively p-bounded operators. We also use well-known methods in the investigation of Sobolev type equations, such as the phase space method, which reduces a singular equation to a regular one, as defined on some subspace of the initial space.

2016, 15(1): 197-217
doi: 10.3934/cpaa.2016.15.197

*+*[Abstract](432)*+*[PDF](485.3KB)**Abstract:**

In this paper we prove the existence and uniqueness of a renormalized solution for nonlinear parabolic equations whose model is \begin{eqnarray} \frac{\partial b(u)}{\partial t} - div\big(a(x,t,u,\nabla u)\big)=f+ div (g), \end{eqnarray} where the right side belongs to $L^{1}(Q)+L^{p'}(0,T;W^{-1,p'}(\Omega))$, where $b(u)$ is a real function of $u$ and where $-div(a(x,t,u,\nabla u))$ is a Leray-Lions type operator with growth $|\nabla u|^{p-1}$ in $\nabla u$, but without any growth assumption on $u$.

2016, 15(1): 219-241
doi: 10.3934/cpaa.2016.15.219

*+*[Abstract](479)*+*[PDF](466.0KB)**Abstract:**

We propose a mathematical analysis of the Swift-Hohenberg equation arising from the phase field theory to model the transition from an unstable to a (meta)stable state. We also consider a recent generalization of the original equation, obtained by introducing an inertial term, to predict fast degrees of freedom in the system. We formulate and prove well-posedness results of the concerned models. Afterwards, we analyse the long-time behavior in terms of global and exponential attractors. Finally, by reading the inertial term as a singular perturbation of the Swift-Hohenberg equation, we construct a family of exponential attractors which is Hölder continuous with respect to the perturbative parameter of the system.

2016, 15(1): 243-260
doi: 10.3934/cpaa.2016.15.243

*+*[Abstract](463)*+*[PDF](489.3KB)**Abstract:**

This paper deals with two-species quasilinear parabolic-parabolic Keller-Segel system $ u_{it}=\nabla\cdot(\phi_i(u_i)\nabla u_i)-\nabla\cdot(\psi_i(u_i)\nabla v)$, $i=1,2$, $v_t=\Delta v-v+u_1+u_2$ in $\Omega\times (0,T)$, subject to the homogeneous Neumann boundary conditions, with bounded domain $\Omega\subset\mathbb{R}^n$, $n\geq2$. We prove that if $\frac{\psi_i(u_i)}{\phi_i(u_i)}\leq C_iu_i^{\alpha_i}$ for $u_i>1$ with $0<\alpha_i<\frac{2}{n}$ and $C_i>0$, $i=1,2$, then the solutions are globally bounded, while if $\frac{\psi_1(u_1)}{\phi_1(u_1)}\geq C_1u_1^{\alpha_1}$ for $u_1>1$ with $\Omega=B_R$, $\alpha_1>\frac{2}{n}$, then for any radial $u_{20}\in C^0(\overline{\Omega})$ and $m_1>0$, there exists positive radial initial data $u_{10}$ with $\int_\Omega u_{10}=m_1$ such that the solution blows up in a finite time $T_{\max}$ in the sense $\lim_{{t\rightarrow T_{\max}}} \|u_1(\cdot,t)+u_2(\cdot,t)\|_{L^{\infty}(\Omega)}=\infty$. In particular, if $\alpha_1>\frac{2}{n}$ with $0<\alpha_2<\frac{2}{n}$, the finite time blow-up for the species $u_1$ is obtained under suitable initial data, a new phenomenon unknown yet even for the semilinear Keller-Segel system of two species.

2016, 15(1): 261-286
doi: 10.3934/cpaa.2016.15.261

*+*[Abstract](388)*+*[PDF](588.0KB)**Abstract:**

As the main problem, the bi-Laplace equation \begin{eqnarray} \Delta^2 u=0 \quad (\Delta=D_x^2+D_y^2) \end{eqnarray} in a bounded domain $\Omega \subset R^2$, with inhomogeneous Dirichlet or Navier-type conditions on the smooth boundary $\partial \Omega$ is considered. In addition, there is a finite collection of curves \begin{eqnarray} \Gamma = \Gamma_1\cup...\cup\Gamma_m \subset \Omega, \end{eqnarray} on which we assume homogeneous Dirichlet conditions $u=0$, focusing at the origin $0 \in \Omega$ (the analysis would be similar for any other point). This makes the above elliptic problem overdetermined. Possible types of the behaviour of solution $u(x,y)$ at the tip $0$ of such admissible multiple cracks, being a singularity point, are described, on the basis of blow-up scaling techniques and spectral theory of pencils of non self-adjoint operators. Typical types of admissible cracks are shown to be governed by nodal sets of a countable family of

*harmonic polynomials*, which are now represented as pencil eigenfunctions, instead of their classical representation via a standard Sturm--Liouville problem. Eventually, for a fixed admissible crack formation at the origin, this allows us to describe

*all*boundary data, which can generate such a blow-up crack structure. In particular, it is shown how the co-dimension of this data set increases with the number of asymptotically straight-line cracks focusing at 0.

2016, 15(1): 287-297
doi: 10.3934/cpaa.2016.15.287

*+*[Abstract](355)*+*[PDF](379.4KB)**Abstract:**

In this paper, using spectral decimation, we prove that the ``hot spots" conjecture holds on higher dimensional Sierpinski gaskets.

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