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Communications on Pure & Applied Analysis

2017 , Volume 16 , Issue 3

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Blowup time and blowup mechanism of small data solutions to general 2-D quasilinear wave equations
Bingbing Ding , Ingo Witt and  Huicheng Yin
2017, 16(3): 719-744 doi: 10.3934/cpaa.2017035 +[Abstract](36) +[HTML](0) +[PDF](544.3KB)
For the 2-D quasilinear wave equation $\sum\nolimits_{i,j = 0}^2 {{g_{ij}}} (\nabla u)\partial _{ij}^2u = 0$, whose coefficients are independent of the solution $u$, the blowup result of small data solution has been established in [1,2] when the null condition does not hold as well as a generic nondegenerate condition of initial data is assumed. In this paper, we are concerned with the more general 2-D quasilinear wave equation $\sum\nolimits_{i,j = 0}^2 {{g_{ij}}} (u,\nabla u)\partial _{ij}^2u = 0$, whose coefficients depend on $u$ and $\nabla u$ simultaneously. When the first weak null condition is not fulfilled and a suitable nondegenerate condition of initial data is assumed, we shall show that the small data smooth solution $u$ blows up in finite time, moreover, an explicit expression of lifespan and blowup mechanism are also established.
On the decay and stability of global solutions to the 3D inhomogeneous MHD system
Junxiong Jia , Jigen Peng and  Kexue Li
2017, 16(3): 745-780 doi: 10.3934/cpaa.2017036 +[Abstract](42) +[HTML](0) +[PDF](473.34KB)
In this paper, we investigative the large time decay and stability to any given global smooth solutions of the 3D incompressible inhomogeneous MHD systems. We prove that given a solution $(a, u, B)$ of (2), the velocity field and the magnetic field decay to zero with an explicit rate, for $u$ which coincide with incompressible inhomogeneous Navier-Stokes equations [1]. In particular, we give the decay rate of higher order derivatives of $u$ and $B$ which are useful to prove our main stability result. For a large solution of (2) denoted by $(a, u, B)$, we show that a small perturbation of the initial data still generates a unique global smooth solution and the smooth solution keeps close to the reference solution $(a, u, B)$. At last, we should mention that the main results in this paper are concerned with large solutions.
Dynamics of a nonlocal dispersal SIS epidemic model
Fei-Ying Yang and  Wan-Tong Li
2017, 16(3): 781-798 doi: 10.3934/cpaa.2017037 +[Abstract](41) +[HTML](3) +[PDF](408.57KB)
This paper is concerned with a nonlocal dispersal susceptible-infected-susceptible (SIS) epidemic model with Dirichlet boundary condition, where the rates of disease transmission and recovery are assumed to be spatially heterogeneous. We introduce a basic reproduction number $R_0$ and establish threshold-type results on the global dynamic in terms of R0. More specifically, we show that if the basic reproduction number is less than one, then the disease will be extinct, and if the basic reproduction number is larger than one, then the disease will persist. Particularly, our results imply that the nonlocal dispersal of the infected individuals may suppress the spread of the disease even though in a high-risk domain.
$W^{1,p}$-estimates for elliptic equations with lower order terms
Byungsoo Kang and  Hyunseok Kim
2017, 16(3): 799-822 doi: 10.3934/cpaa.2017038 +[Abstract](38) +[HTML](0) +[PDF](491.2KB)
We consider the Neumann and Dirichlet problems for second-order linear elliptic equations $-\textrm{div} \, (A \nabla u) -b \cdot \nabla u + \lambda u = f + \mbox{div} \, F, \quad $ $-\textrm{div} \, (A^t \nabla v) + \textrm{div} \, (v b) + \lambda v = g + \mbox{div} \, G $ in a bounded Lipschitz domain $\Omega \subset \mathbb{R}^n$, $n \geq 2$, where $A: \mathbb{R}^n \to \mathbb{R}^{n^2}$, $b: \Omega \to \mathbb{R}^n$ and $\lambda \geq 0$ are given. Some $W^{1, 2}$-estimates have been already known, provided that $A \in L^\infty(\Omega)^{n^2}$ and $b \in L^r(\Omega)^n$, where $n \leq r < \infty$ if $n \geq 3$ and $2 < r < \infty$ if $n=2$. Under more regularity assumptions on $A$ and $\Omega$, we establish the existence and uniqueness of weak solutions satisfying $W^{1, p}$-estimates. Our $W^{1, p}$-estimates are uniform on $\lambda \geq 0$ for the case of the Dirichlet problems. For the Neumann problems, the $W^{1, p}$-estimates are uniform with respect to $\lambda \geq 0$ if $f$ and $g$ satisfy some compatibility conditions. These uniform estimates allow us to obtain strong stability results in $W^{1, p}$ with respect to $\lambda $ for the Neumann and Dirichlet problems.
Infinitely many solutions for a class of perturbed elliptic equations with nonlocal operators
Liang Zhang , X. H. Tang and  Yi Chen
2017, 16(3): 823-842 doi: 10.3934/cpaa.2017039 +[Abstract](33) +[HTML](1) +[PDF](508.2KB)
In this paper, we consider the following perturbed nonlocal elliptic equation where $\Omega$ is a smooth bounded domain in $\mathbb{R}{^N}$, $\lambda$ is a real parameter and $g$ is a non-odd perturbation term. If $f$ is odd in $u$ and satisfies various superlinear growth conditions at infinity in $u$, infinitely many solutions are obtained in spite of the lack of the symmetry of this problem for any $\lambda\in \mathbb{R}$. The results obtained in this paper may be seen as natural extensions of some classical theorems to the case of nonlocal operators. Moreover, the methods used in this paper can be also applied to obtain some new results for the classical Laplace equation with Dirichlet boundary conditions.
On nonexistence of solutions to some nonlinear parabolic inequalities
Olga Salieva
2017, 16(3): 843-853 doi: 10.3934/cpaa.2017040 +[Abstract](94) +[HTML](0) +[PDF](333.08KB)
We obtain sufficient conditions for nonexistence of positive solutions to some nonlinear parabolic inequalities with coefficients possessing singularities on unbounded sets.
On weighted mixed-norm Sobolev estimates for some basic parabolic equations
Li Ping , Pablo Raúl Stinga and  José Luis Torrea
2017, 16(3): 855-882 doi: 10.3934/cpaa.2017041 +[Abstract](37) +[HTML](0) +[PDF](526.8KB)
Novel global weighted parabolic Sobolev estimates, weighted mixed-norm estimates and a.e. convergence results of singular integrals for evolution equations are obtained. Our results include the classical heat equation \begin{eqnarray} \partial_tu=\Delta u+f, \end{eqnarray} the harmonic oscillator evolution equation \begin{eqnarray} \partial_tu=\Delta u-|x|^2u+f, \end{eqnarray} and their corresponding Cauchy problems. We also show weighted mixed-norm estimates for solutions to degenerate parabolic extension problems arising in connection with the fractional space-time nonlocal equations $(\partial_t-\Delta)^s u=f$ and $(\partial_t-\Delta+|x|^2)^s u=f$, for $0 < s < 1$.
$L^p$ boundedness for maximal functions associated with multi-linear pseudo-differential operators
Jiao Chen , Wei Dai and  Guozhen Lu
2017, 16(3): 883-898 doi: 10.3934/cpaa.2017042 +[Abstract](38) +[HTML](0) +[PDF](467.92KB)
In this paper, we establish the $L^p$ estimates for the maximal functions associated with the multilinear pseudo-differential operators. Our main result is Theorem 1.2. There are several major different ingredients and extra difficulties in our proof from those in Grafakos, Honzík and Seeger [15] and Honzík [22] for maximal functions generated by multipliers. First, in order to eliminate the variable $x$ in the symbols, we adapt a non-smooth modification of the smooth localization method developed by Muscalu in [26,30]. Then, by applying the inhomogeneous Littlewood-Paley dyadic decomposition and a discretization procedure, we can reduce the proof of Theorem 1.2 into proving the localized estimates for localized maximal functions generated by discrete paraproducts. The non-smooth cut-off functions in the localization procedure will be essential in establishing localized estimates. Finally, by proving a key localized square function estimate (Lemma 4.3) and applying the good-$\lambda$ inequality, we can derive the desired localized estimates.
Weighted Lorentz estimates for nondivergence linear elliptic equations with partially BMO coefficients
Junjie Zhang and  Shenzhou Zheng
2017, 16(3): 899-914 doi: 10.3934/cpaa.2017043 +[Abstract](23) +[HTML](0) +[PDF](436.04KB)
We prove weighted Lorentz estimates of the Hessian of strong solution for nondivergence linear elliptic equations $a_{ij}(x)D_{ij}u(x)=f(x)$. The leading coefficients are assumed to be measurable with respect to one variable and have small BMO semi-norms with respect to the other variables. Here, an approximation method, Lorentz boundedness of the Hardy-Littlewood maximal operators and an equivalent representation of Lorentz norm are employed.
Tug-of-war games with varying probabilities and the normalized $p(x)$-Laplacian
Ángel Arroyo , Joonas Heino and  Mikko Parviainen
2017, 16(3): 915-944 doi: 10.3934/cpaa.2017044 +[Abstract](44) +[HTML](0) +[PDF](505.66KB)
We study a two player zero-sum tug-of-war game with varying probabilities that depend on the game location x. In particular, we show that the value of the game is locally asymptotically Hölder continuous. The main difficulty is the loss of translation invariance. We also show the existence and uniqueness of values of the game. As an application, we prove that the value function of the game converges to a viscosity solution of the normalized p(x) -Laplacian.
Non-collapsing for a fully nonlinear inverse curvature flow
Yannan Liu and  Hongjie Ju
2017, 16(3): 945-952 doi: 10.3934/cpaa.2017045 +[Abstract](26) +[HTML](0) +[PDF](311.98KB)
In this paper, we study a fully nonlinear inverse curvature flow in Euclidean space, and prove a non-collapsing property for this flow using maximum principle. Precisely, we show that upon some conditions on speed function, the curvature of the largest touching interior ball is bounded by a multiple of the speed.
Multiple positive standing wave solutions for Schrödinger equations with oscillating state-dependent potentials
Renata Bunoiu , Radu Precup and  Csaba Varga
2017, 16(3): 953-972 doi: 10.3934/cpaa.2017046 +[Abstract](29) +[HTML](0) +[PDF](460.56KB)
Motivated by relevant physical applications, we study Schrödinger equations with state-dependent potentials. Existence, localization and multiplicity results are established for positive standing wave solutions in the case of oscillating potentials. To this aim, a localized Pucci-Serrin type critical point theorem is first obtained. Two examples are then given to illustrate the new theory.
Equivalence of sharp Trudinger-Moser-Adams inequalities
Nguyen Lam
2017, 16(3): 973-998 doi: 10.3934/cpaa.2017047 +[Abstract](39) +[HTML](0) +[PDF](462.6KB)
Improved Trudinger-Moser-Adams type inequalities in the spirit of Lions were recently studied in [21]. The main purpose of this paper is to prove the equivalence of these versions of the Trudinger-Moser-Adams type inequalities and to set up the relations of these Trudinger-Moser-Adams best constants. Moreover, using these identities, we will investigate the existence and nonexistence of the optimizers for some Trudinger-Moser-Adams type ineq
Positive ground state solutions of a quadratically coupled Schrödinger system
Zhanping Liang , Yuanmin Song and  Fuyi Li
2017, 16(3): 999-1012 doi: 10.3934/cpaa.2017048 +[Abstract](34) +[HTML](0) +[PDF](389.02KB)
In this paper, we study the following quadratically coupled Schrödinger system: where $\Omega\subset\mathbb{R}^6$ is a smooth bounded domain, $-\lambda (\Omega) < \lambda_1, \lambda_2 < 0, \mu_1, \mu_2, \alpha, \gamma>0$, and $\lambda (\Omega)$ is the first eigenvalue of $-\Delta$ with the Dirichlet boundary condition. The main difficulty to investigate this kind of equations is caused by the fact that all the quadratic nonlinearities, including the coupling terms, are of critical growth. By the methods used in [Zhenyu Guo, Positive ground state solutions of a nonlinearly coupled Schrödinger system with critical exponents in [Zhenyu Guo, Positive ground state solutions of a nonlinearly coupled Schrödinger system with critical exponents in $\mathbb{R}^4$, J. Math. Anal. Appl., 430(2):950-970, 2015], the existence of positive ground state solutions of the system is established with more ingenious hypotheses.
Boundary layer analysis from the Keller-Segel system to the aggregation system in one space dimension
Jiahang Che , Li Chen , Simone Göttlich , Anamika Pandey and  Jing Wang
2017, 16(3): 1013-1036 doi: 10.3934/cpaa.2017049 +[Abstract](224) +[HTML](0) +[PDF](1098.3KB)
This article is devoted to the discussion of the boundary layer which arises from the one-dimensional parabolic elliptic type Keller-Segel system to the corresponding aggregation system on the half space case. The characteristic boundary layer is shown to be stable for small diffusion coefficients by using asymptotic analysis and detailed error estimates between the Keller-Segel solution and the approximate solution. In the end, numerical simulations for this boundary layer problem are provided.
Global existence of solutions to an attraction-repulsion chemotaxis model with growth
Sainan Wu , Junping Shi and  Boying Wu
2017, 16(3): 1037-1058 doi: 10.3934/cpaa.2017050 +[Abstract](24) +[HTML](0) +[PDF](499.29KB)
An attraction-repulsion chemotaxis model with nonlinear chemotactic sensitivity functions and growth source is considered. The global-in-time existence and boundedness of solutions are proved under some conditions on the nonlinear sensitivity functions and growth source function. Our results improve the earlier ones for the linear sensitivity functions.
Stabilizing blow up solutions to nonlinear Schrödinger equations
Laurent Di Menza and  Olivier Goubet
2017, 16(3): 1059-1082 doi: 10.3934/cpaa.2017051 +[Abstract](30) +[HTML](0) +[PDF](964.18KB)
In this paper, we consider the critical nonlinear Schrödinger equations in ${\mathbb{R}^2}$ with an oscillating nonlinearity, in a radial geometry. We numerically investigate the influence of the oscillations on the time of existence for the corresponding solution, on the spirit of the recent result of Cazenave and Scialom. It can be observed that the solution converges to the solution of a limit equation obtained with the weak limit of the oscillatory term, starting either with Gaussian data as well as standing waves solutions.
Multiple positive solutions of a Sturm-Liouville boundary value problem with conflicting nonlinearities
Guglielmo Feltrin
2017, 16(3): 1083-1102 doi: 10.3934/cpaa.2017052 +[Abstract](26) +[HTML](0) +[PDF](647.2KB)
We study the second order nonlinear differential equation $u'' + \sum\limits_{i = 1}^m {} {\alpha _i}{a_i}(x){g_i}(u) - \sum\limits_{j = 1}^{m + 1} {} {\beta _j}{b_j}(x){k_j}(u) = 0,{\rm{ }}$ where $\alpha_{i}, \beta_{j}>0$, $a_{i}(x), b_{j}(x)$ are non-negative Lebesgue integrable functions defined in $\mathopen{[}0, L\mathclose{]}$, and the nonlinearities $g_{i}(s), k_{j}(s)$ are continuous, positive and satisfy suitable growth conditions, as to cover the classical superlinear equation $u"+a(x)u.{p} = 0$, with $p>1$.When the positive parameters $\beta_{j}$ are sufficiently large, we prove the existence of at least $2.{m}-1$positive solutions for the Sturm-Liouville boundary value problems associated with the equation.The proof is based on the Leray-Schauder topological degree for locally compact operators on open and possibly unbounded sets.Finally, we deal with radially symmetric positive solutions for the Dirichlet problems associated with elliptic PDEs.

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