American Institute of Mathematical Sciences

In this paper, we establish the existence of random attractors for stochastic parabolic equations driven by additive noise as well as deterministic non-autonomous forcing terms in weighted Lebesgue spaces \begin{document}$ L_{\delta}^r(\mathcal{O})$\end{document}, where \begin{document}$ 1<r<\infty ,\ \delta$\end{document} is the distance from \begin{document}$ x$\end{document} to the boundary. The nonlinearity \begin{document}$ f(x,u)$\end{document} of equation depending on the spatial variable does not have the bound on the derivative in \begin{document}$ u$\end{document}, and then causes critical exponent. In both subcritical and critical cases, we get the well-posedness and dissipativeness of the problem under consideration and, by smoothing property of heat semigroup in weighted space, the asymptotical compactness of random dynamical system corresponding to the original system.

We study a stochastic parabolic evolution equation of the form \begin{document}$ dX+AXdt = F(t)dt+G(t)dW(t)$ \end{document} in Banach spaces. Existence of mild and strict solutions and their space-time regularity are shown in both the deterministic and stochastic cases. Abstract results are applied to a nonlinear stochastic heat equation.

We study symmetry and asymptotic behavior of ground state solutions for the doubly coupled nonlinear Schrödinger elliptic system

\begin{document}$\left\{ {\begin{array}{*{20}{l}}{ - \Delta u + {\lambda _1}u + \kappa v = {\mu _1}{u^3} + \beta u{v^2}}&{\quad {\rm{ in}}\;\;\Omega ,}\\{ - \Delta v + {\lambda _2}v + \kappa u = {\mu _2}{v^3} + \beta {u^2}v}&{\quad {\rm{ in}}\;\;\Omega ,}\\{u = v = 0\;on\;\;\partial \Omega \;({\rm{or}}\;u,v \in {H^1}({\mathbb{R}^N})\;{\rm{as}}\;\Omega = {\mathbb{R}^N}),}&{}\end{array}} \right.$ \end{document}

where \begin{document}$ N≤3, Ω\subseteq\mathbb{R}^N$\end{document} is a smooth domain. First we establish the symmetry of ground state solutions, that is, when \begin{document}$ Ω$\end{document} is radially symmetric, we show that ground state solution is foliated Schwarz symmetric with respect to the same point. Moreover, ground state solutions must be radially symmetric under the condition that \begin{document}$ Ω$\end{document} is a ball or the whole space \begin{document}$ \mathbb{R}^N$\end{document}. Next we investigate the asymptotic behavior of positive ground state solution as \begin{document}$ κ\to 0^-$\end{document}, which shows that the limiting profile is exactly a minimizer for \begin{document}$ c_0$\end{document} (the minimized energy constrained on Nehari manifold corresponds to the limit systems). Finally for a system with three equations, we prove the existence of ground state solutions whose all components are nonzero.

In this paper, we consider the nonnegative solutions of the following system of integral form:

\begin{document}$\left\{ \begin{matrix} {{u}_{i}}(x)=\int_{{{\mathbf{R}}^{n}}}{\frac{1}{|x-y{{|}^{n-\alpha }}}}{{f}_{i}}(u(y))dy,\ \ x\in {{\mathbf{R}}^{n}},\ \ i=1,\cdots ,m, \\ 0<\alpha <n,\text{and }\ u(x)=({{u}_{1}}(x),{{u}_{2}}(x),\cdots ,{{u}_{m}}(x)). \\\end{matrix} \right.\ \ \ \ \ \ \left( 1 \right)$ \end{document}

Here \begin{document}$ f_i(u)∈ C^1(\mathbf{R^m_+})\bigcap$\end{document}\begin{document}$ C^0(\mathbf{\overline{R^m_+}})(i = 1,2,···,m)$\end{document} are real-valued functions, nonnegative, homogeneous of degree \begin{document}$ β_{i}$\end{document}, where \begin{document}$ 0<β_{i} ≤q \frac{n+α}{n-α}$\end{document}, and monotone nondecreasing with respect to the variables \begin{document}$ u_1, u_2, ···, u_m$\end{document}. We show that the nonnegative solution \begin{document}$ u = (u_1,u_2,···,u_m)$\end{document} is radially symmetric in the critical and subcritical case by method of moving planes in an integral form and \begin{document}$ u$\end{document} must be zero in the subcritical case.

Futhermore, we consider the form of \begin{document}$ f_i(u) = \sum_{r = 1}^{k}f_{ir}(u),$\end{document} where \begin{document}$ f_{ir}(u)$\end{document} are real-valued homogeneous functions of various degrees \begin{document}$ β_{ir}, r = 1,2,···,k$\end{document} and \begin{document}$ 0 <β_{ir} ≤q \frac{n+α}{n-α}$\end{document}. We also show that the radial symmetry property of the nonnegative solution. Due to the homogeneous of degree can be different, the more intricate method is needed to deal with this difficulty.

In this paper, we investigate the soliton dynamics under slowly varying medium for the BBM equation, that is how the solution of this equation evolve when the time goes. We construct the approximate solution of this equation and prove that the error term due to the approximate solution can be controlled. By using the method of Lyapunov and Weinstein functions, we prove that the approximate solution is stable.

In this paper the fractional Q-curvature problem on three dimensional CR sphere is considered. By using the critical points theory at infinity, an existence result is obtained.

We consider the fourth order problem \begin{document} $Δ^{2}u = λ f(u)$ \end{document} on a general bounded domain \begin{document} $Ω$ \end{document} in \begin{document} $R^{n}$ \end{document} with the Navier boundary condition \begin{document} $u = Δ u = 0$ \end{document} on \begin{document} $\partial Ω$ \end{document}. Here, \begin{document} $λ$ \end{document} is a positive parameter and \begin{document} $ f:[0, a_{f}) \to \Bbb{R}_{+} $ \end{document} \begin{document} $ \left( {0 < {a_f} \le \infty } \right)$ \end{document} is a smooth, increasing, convex nonlinearity such that \begin{document} $ f(0) > 0 $ \end{document} and which blows up at \begin{document} $ {a_f} $ \end{document}. Let

\begin{document}$0<τ_{-}: = \liminf\limits_{t \to a_{f}} \frac{f(t)f''(t)}{f'(t)^{2}}≤q τ_{+}: = \limsup\limits_{t \to a_{f}} \frac{f(t)f''(t)}{f'(t)^{2}}<2.$ \end{document}

We show that if $u_{m}$ is a sequence of semistable solutions correspond to $λ_{m}$ satisfy the stability inequality

\begin{document}$\sqrt{λ_{m}}\int{{_{Ω}}}\sqrt{f'(u_{m})}\phi ^{2}dx≤\int{{_{Ω}}}|\nablaφ|^{2}dx, ~~\text{for all}~\phi ∈ H^{1}_{0}(Ω), $ \end{document}

then $\sup_{m} ||u_{m}||_{L^{∞}(Ω)}<a_{f}$ for $n< \frac{4α_{*}(2-τ_{+})+2τ_{+}}{τ_{+}}\max \{1, τ_{+}\}, $ where $α^{*}$ is the largest root of the equation

\begin{document}$(2-τ_{-})^{2} α^{4}- 8(2-τ_{+})α^{2}+4(4-3τ_{+})α-4(1-τ_{+}) = 0.$ \end{document}

In particular, if $τ_{-} = τ_{+}: = τ$, then $\sup_{m} ||u_{m}||_{L^{∞}(Ω)}<a_{f}$ for $n≤12$ when $τ≤ 1$, and for $n≤7$ when $τ≤ 1.57863$. These estimates lead to the regularity of the corresponding extremal solution $u^{*}(x) = \lim_{λ\uparrowλ^{*}}u_{λ}(x), $ where $λ^*$ is the extremal parameter of the eigenvalue problem.

Any positive power of the Laplacian is related via its Fourier symbol to a hypersingular integral with finite differences. We show how this yields a pointwise evaluation which is more flexible than other notions used so far in the literature for powers larger than 1; in particular, this evaluation can be applied to more general boundary value problems and we exhibit explicit examples. We also provide a natural variational framework and, using an asymptotic analysis, we prove how these hypersingular integrals reduce to polyharmonic operators in some cases. Our presentation aims to be as self-contained as possible and relies on elementary pointwise calculations and known identities for special functions.

In this paper, we study the initial-boundary value problems of the coupled modified Korteweg-de Vries equation formulated on the finite interval with Lax pairs involving \begin{document}$3× 3$\end{document} matrices via the Fokas method. We write the solution in terms of the solution of a \begin{document}$3× 3$\end{document} Riemann-Hilbert problem. The relevant jump matrices are explicitly expressed in terms of the three matrix-value spectral functions \begin{document}$s(k)$\end{document}, \begin{document}$S(k)$\end{document}, and \begin{document}$S_{L}(k)$\end{document}, which are determined by the initial values, boundary values at \begin{document}$x = 0$\end{document}, and at \begin{document}$x = L$\end{document}, respectively. Some of the boundary values are known for a well-posed problem, however, the remaining boundary data are unknown. By using the so-called global relation, the unknown boundary values can be expressed in terms of the given initial and boundary data via a Gelfand-Levitan-Marchenko representation.

We establish the existence and stability of smooth large-amplitude traveling waves to nonlinear conservation laws modeling image processing with general flux functions. We innovatively construct a weight function in the weighted energy estimates to overcome the difficulties caused by the absence of the convexity of fluxes in our model. Moreover, we prove that if the integral of the initial perturbation decays algebraically or exponentially in space, the solution converges to the traveling waves with rates in time, respectively. Furthermore, we are able to construct another new weight function to deal with the degeneracy of fluxes in establishing the stability.

We consider the Cauchy problem for nonlinear abstract wave equations in a Hilbert space. Our main goal is to show that this problem has solutions with arbitrary positive initial energy that blow up in a finite time. The main theorem is proved by employing a result on growth of solutions of abstract nonlinear wave equation and the concavity method. A number of examples of nonlinear wave equations are given. A result on blow up of solutions with arbitrary positive initial energy to the initial boundary value problem for the wave equation under nonlinear boundary conditions is also obtained.

In this paper we discuss a family of viscous Cahn-Hilliard equations with a non-smooth viscosity term. This system may be viewed as an approximation of a ''forward-backward'' parabolic equation. The resulting problem is highly nonlinear, coupling in the same equation two nonlinearities with the diffusion term. In particular, we prove existence of solutions for the related initial and boundary value problem. Under suitable assumptions, we also state uniqueness and continuous dependence on data.

We consider the equation

\begin{document}$-{y}''(x)+q(x)y(x)=f(x),\ \ \ \ x\in \mathbb{R}\text{ }\ \ \ \ \ \ \ \ \ \ \left( 1 \right)$ \end{document}

where \begin{document}$f∈ L_p^{\text{loc}}(\mathbb R),$\end{document} \begin{document}$p∈[1,∞)$\end{document} and \begin{document}$0≤ q∈ L_1^{\text{loc}}(\mathbb R).$\end{document} By a solution of (1) we mean any function \begin{document}$y,$\end{document} absolutely continuous together with its derivative and satisfying (1) almost everywhere in \begin{document}$\mathbb R.$\end{document} Let positive and continuous functions \begin{document}$μ(x)$\end{document} and \begin{document}$θ(x)$\end{document} for \begin{document}$x∈\mathbb R$\end{document} be given. Let us introduce the spaces

\begin{document}$\begin{align} & {{L}_{p}}(\mathbb{R},\mu )=\left\{ f\in L_{p}^{\text{loc}}(\mathbb{R}):\|f\|_{{{L}_{p}}(\mathbb{R},\mu )}^{p}=\int_{-\infty }^{\infty }{|}\mu (x)f(x){{|}^{p}}dx < \infty \right\}, \\ & {{L}_{p}}(\mathbb{R},\theta )=\left\{ f\in L_{p}^{\text{loc}}(\mathbb{R}):\|f\|_{{{L}_{p}}(\mathbb{R},\theta )}^{p}=\int_{-\infty }^{\infty }{|}\theta (x)f(x){{|}^{p}}dx <\infty \right\}. \\ \end{align}$ \end{document}

In the present paper, we obtain requirements to the functions \begin{document}$μ,θ$\end{document} and \begin{document}$q$\end{document} under which

1) for every function \begin{document}$f∈ L_p(\mathbb R,θ)$\end{document} there exists a unique solution (1) \begin{document}$y∈ L_p(\mathbb R,μ)$\end{document} of (1);

2) there is an absolute constant \begin{document}$c(p)∈(0,∞)$\end{document} such that regardless of the choice of a function \begin{document}$f∈ L_p(\mathbb R,θ)$\end{document} the solution of (1) satisfies the inequality

\begin{document}$\|y\|_{L_p(\mathbb R,μ)}≤ c(p)\|f\|_{L_p(\mathbb R,θ)}.$ \end{document}

We consider the following fractional Hénonsystem

\begin{document}$\left\{ \begin{array}{*{35}{l}} {}&{{(-\vartriangle )}^{\alpha /2}}u = |x{{|}^{a}}{{v}^{p}},~~&x\in {{R}^{n}}, \\ {}&{{(-\vartriangle )}^{\alpha /2}}v = |x{{|}^{b}}{{u}^{q}},~~&x\in {{R}^{n}}, \\ {}&u\ge 0,v\ge 0,&{} \\\end{array} \right.$ \end{document}

for \begin{document}$0<α<2$\end{document} and \begin{document}$a, b$\end{document} \begin{document}$≥0$\end{document}, \begin{document}$n≥2$\end{document}. Under rather weaker assumptions, by using a direct method of moving planes, we prove the nonexistence and symmetry of positive solutions in the subcritical case where \begin{document}$1<p<\frac{n+α+a}{n-α}$\end{document} and \begin{document}$1<q<\frac{n+α+b}{n-α}$\end{document}.

We obtain global well-posedness, scattering, and global \begin{document}$L_t^4H_{x}^{1,4}$\end{document} spacetime bounds for energy-space solutions to the energy-subcritical nonlinear Schrödinger equation

\begin{document}$iu_t+Δ u = u(e^{4π |u|^2}-1)$ \end{document}

in two spatial dimensions. Our approach is perturbative; we view our problem as a perturbation of the mass-critical NLS to employ the techniques of Tao-Visan-Zhang from [25]. This permits us to combine the known spacetime estimates for mass-critical NLS proved by Dodson [12] and the work of [15] and [14] to prove corresponding spacetime estimates which imply scattering.

The aim of this paper is to study the following problem

\begin{document}$(P_{\lambda}) \equiv\left\{\begin{array}{rcll}(-\Delta)^s u& = &\lambda u^{q}+u^{p}&{\text{ in }}\Omega,\\ u&>&0 &{\text{ in }} \Omega, \\ \mathcal{B}_{s}u& = &0 &{\text{ in }} \mathbb{R}^{N}\backslash \Omega,\end{array}\right.$ \end{document}

with \begin{document}$0<q<1<p$\end{document}, \begin{document}$N>2s$\end{document}, \begin{document}$λ> 0$\end{document}, \begin{document}$Ω \subset \mathbb{R}^{N}$\end{document} is a smooth bounded domain,

\begin{document}$(-Δ)^su(x) = a_{N,s}\;P.V.∈t_{\mathbb{R}^{N}}\frac{u(x)-u(y)}{|x-y|^{N+2s}}\,dy,$ \end{document}

\begin{document}$a_{N,s}$\end{document} is a normalizing constant, and \begin{document}$\mathcal{B}_{s}u = uχ_{Σ_{1}}+\mathcal{N}_{s}uχ_{Σ_{2}}.$\end{document} Here, \begin{document}$Σ_{1}$\end{document} and \begin{document}$Σ_{2}$\end{document} are open sets in \begin{document}$\mathbb{R}^{N}\backslash Ω$\end{document} such that \begin{document}$Σ_{1} \cap Σ_{2} = \emptyset$\end{document} and \begin{document}$\overline{Σ}_{1}\cup \overline{Σ}_{2} = \mathbb{R}^{N}\backslash Ω.$\end{document}

In this setting, \begin{document}$\mathcal{N}_{s}u$\end{document} can be seen as a Neumann condition of nonlocal type that is compatible with the probabilistic interpretation of the fractional Laplacian, as introduced in [20], and \begin{document}$\mathcal{B}_{s}u$\end{document} is a mixed Dirichlet-Neumann exterior datum. The main purpose of this work is to prove existence, nonexistence and multiplicity of positive energy solutions to problem (\begin{document}$P_{λ}$\end{document}) for suitable ranges of \begin{document}$λ$\end{document} and \begin{document}$p$\end{document} and to understand the interaction between the concave-convex nonlinearity and the Dirichlet-Neumann data.

In this paper, we are concerned with the existence of ground state solutions for the following quasilinear Schrödinger equation:

\begin{document}$-Δ u+V(x)u-Δ (u^2)u = K(x)|u|^{22^*-2}u+g(x,u), \ \ x∈ \mathbb{R}^N\ \ \ \ \ \ \ \ \ \ \left( 1 \right)$ \end{document}

where $N≥ 3$, $V, \ g$ are asymptotically periodic functions in $x$. By combining variational methods and the concentration-compactness principle, we obtain a ground state solution for equation (1) under a new reformative condition which unify the asymptotic processes of $V, g $ at infinity.

In this article, we study the existence of sign-changing solutions for a problem driven by a non-local integrodifferential operator with homogeneous Dirichlet boundary condition

\begin{document}$\left\{ \begin{array}{ll}-\mathcal{L}_Ku = f(x,u) &\text{in}~Ω, \\u = 0 &\text{in}~\mathbb{R}^n\setminusΩ, \end{array} \right.\ \ \ \ \ \ \ \ \ \left( 1 \right)$ \end{document}

where $Ω\subset\mathbb{R}^n(n≥2)$ is a bounded, smooth domain and $f(x, u)$ is asymptotically linear at infinity with respect to $u$. By introducing some new ideas and combining constraint variational method with the quantitative deformation lemma, we prove that there exists a sign-changing solution of problem (1).

We study the short and large time behaviour of solutions of nonlocal heat equations of the form $\partial_tu+\mathcal{L} u = 0$. Here $\mathcal{L}$ is an integral operator given by a symmetric nonnegative kernel of Lévy type, that includes bounded and unbounded transition probability densities. We characterize when a regularizing effect occurs for small times and obtain $L^q$-$L^p$ decay estimates, $1≤ q < p < ∞$ when the time is large. These properties turn out to depend only on the behaviour of the kernel at the origin or at infinity, respectively, without need of any information at the other end. An equivalence between the decay and a restricted Nash inequality is shown. Finally we deal with the decay of nonlinear nonlocal equations of porous medium type $\partial_tu+\mathcal{L}Φ(u) = 0$.

The existence of traveling wave solutions and wave train solutions of a diffusive ratio-dependent predator-prey system with distributed delay is proved. For the case without distributed delay, we first establish the existence of traveling wave solution by using the upper and lower solutions method. Second, we prove the existence of periodic traveling wave train by using the Hopf bifurcation theorem. For the case with distributed delay, we obtain the existence of traveling wave and traveling wave train solutions when the mean delay is sufficiently small via the geometric singular perturbation theory. Our results provide theoretical basis for biological invasion of predator species.

In this article, we prove the existence, multiplicity and concentration of non-trivial solutions for the following indefinite fractional elliptic equation with concave-convex nonlinearities:

\begin{document}$\left\{\begin{array}{*{20}{l}}(-Δ)^α u+V_λ(x)u = a(x)|u|^{q-2}u+b(x)|u|^{p-2}u &{\rm in}\,\,\mathbb{R}^N,\\u≥0\,\,&{\rm in}\,\,\mathbb{R}^N, \end{array} \right.$ \end{document}

where $0<α<1$, $N>2α$, $1<q<2<p<2^*_α$ with $ 2^*_α = 2N/(N-2α)$, the potential $V_λ(x) = λ V^+(x)-V^-(x)$ with $V^± = \max\{± V, 0\}$ and the parameter $λ>0$. Our multiplicity results are based on studying the decomposition of the Nehari manifold.

We study in detail the existence, nonexistence and behavior of global minimizers, ground states and corresponding energy levels of the \begin{document}$(p, q)$\end{document} -Laplace equation \begin{document}$-Δ_p u -Δ_q u = α |u|^{p-2}u + β |u|^{q-2}u$\end{document} in a bounded domain $Ω \subset \mathbb{R}^N$ under zero Dirichlet boundary condition, where \begin{document}$p > q > 1$\end{document} and \begin{document}$α, β ∈ \mathbb{R}$\end{document} . A curve on the \begin{document}$(α, β)$\end{document} -plane which allocates a set of the existence of ground states and the multiplicity of positive solutions is constructed. Additionally, we show that eigenfunctions of the p-and q-Laplacians under zero Dirichlet boundary condition are linearly independent.

We prove the existence of a loop type component of non-negative solutions for an indefinite elliptic equation with a homogeneous Neumann boundary condition. This result complements our previous results obtained in [12], where the existence of another loop type component was established in a different situation. Our proof combines local and global bifurcation theory, rescaling and regularizing arguments, a priori bounds, and Whyburn's topological method. A further investigation of the loop type component established in [12] is also provided.

In this paper, we discuss exact multiplicity and bifurcation curves of positive solutions of the one-dimensional Minkowski-curvature problem

\begin{document}$\begin{equation*}\left\{\begin{array}{l}\left[ -u^{\prime }/\sqrt{1-u^{\prime 2}}\right] ^{\prime }=\lambda f(u),\,\,\,\,\,\,-L < x < L, \\u(-L)=u(L)=0,\end{array}\right.\end{equation*} $ \end{document}

where $λ >0$ is a bifurcation parameter, $L>0$ is an evolution parameter, \begin{document}$f∈ C[0, ∞)\cap C^{2}(0, ∞), $\end{document} \begin{document}$f(u)>0$\end{document} for \begin{document}$u>0$\end{document} , and \begin{document}$f^{\prime \prime }(u)$\end{document} is not sign-changing on \begin{document}$\left( 0,\infty \right)$\end{document} .We find that if \begin{document}$f^{\prime \prime }(u)≤q 0$\end{document} for \begin{document}$u>0$\end{document} , the shapes of bifurcation curves are monotone increasing for $L>0$, and if \begin{document}$f^{\prime \prime }(u)>0$\end{document} for \begin{document}$u>0$\end{document} and \begin{document}$f(u)$\end{document} satisfies some suitable hypotheses, the shapes of bifurcation curves has three possibilities. Furthermore, we study, in the \begin{document}$(\lambda ,L,{\left\| u \right\|_\infty })$\end{document} -space, the shapes and structures of the bifurcation surfaces. Finally, we give an application for this problem with a nonlinear term \begin{document}$f(u) = u^{p}+u^{q}$\end{document} where \begin{document}$q≥p>0$\end{document} satisfy some conditions.

In this paper we study the positive solutions to the \begin{document} $n\times n$ \end{document} \begin{document} $p$ \end{document}-Laplacian system:

\begin{document}$\begin{equation*}\begin{cases}-\left(\varphi_{p_1}(u_1')\right)' = \lambda h_1(t) \left(u_1^{p_1-1-\alpha_1}+f_1(u_2)\right),\quad t\in (0,1),\\-\left(\varphi_{p_2}(u_2')\right)' = \lambda h_2(t) \left(u_2^{p_2-1-\alpha_2}+f_2(u_3)\right),\quad t\in (0,1),\\\quad\quad\quad\vdots\qquad\,\: =\quad\quad\quad\quad\quad\quad \vdots\\-\left(\varphi_{p_n}(u_n')\right)' = \lambda h_n(t) \left(u_n^{p_n-1-\alpha_n}+f_n(u_1)\right),~~\, t\in (0,1),\\\quad\,\,\,\, u_j(0)=0=u_j(1); ~~ j=1,2,\dots,n, \\ \end{cases}\end{equation*}$ \end{document}

where \begin{document} $\lambda$ \end{document} is a positive parameter, \begin{document} $p_j>1$ \end{document}, \begin{document} $\alpha_j\in(0,p_j-1)$ \end{document}, \begin{document} $\varphi_{p_j}(w)=|w|^{p_j-2}w$ \end{document}, and \begin{document} $h_j \in C((0,1),(0, \infty))\cap L^1((0,1),(0,\infty))$ \end{document} for \begin{document} $j=1,2,\dots,n$ \end{document}. Here \begin{document} $f_j:[0,\infty)\rightarrow[0,\infty)$ \end{document}, \begin{document} $j=1,2,\dots,n$ \end{document} are nontrivial nondecreasing continuous functions with \begin{document} $f_j(0)=0$ \end{document} and satisfy a combined sublinear condition at infinity. We discuss here a bifurcation result, an existence result for \begin{document} $\lambda>0$ \end{document}, and a multiplicity result for a certain range of \begin{document} $\lambda$ \end{document}. We establish our results through the method of sub-super solutions.

In this paper we finish the study of the cyclicity ( i.e. the maximum number of limit cycles) of the degenerate graphic \begin{document} $DF_{2a}$ \end{document} of [6] which is initiated in [5]. More precisely, we prove that the graphic \begin{document} $DF_{2a}$ \end{document} has a finite cyclicity. The goal of the program [6] is to solve the finiteness part of Hilbert's 16th problem for quadratic polynomial systems. We use techniques from geometric singular perturbation theory, including the family blow-up.