For Sobolev exponents greater than $3/2$, it is proved that the data-to-solution map for the $b$-family equation is continuous from $H^s({\mathbb{T}})$ into $C([0,T]; H^s({\mathbb{T}}))$ but not uniformly continuous. The proof of non-uniform dependence on initial data is based on approximate solutions and delicate commutator and multiplier estimates. The novelty of the proof lies in the fact that it makes no use of conserved quantities. Furthermore, it is shown that the solution map is Hölder continuous in a weaker topology.

We consider a coupled system of Schrödinger equations with time-periodic coefficients \begin{eqnarray*} i\, u_t = -\Delta u + V(x,t)u + g(x,t)v \\ i\, v_t = -\Delta v + W(x,t)v + g(x,t)u \end{eqnarray*} on the Hilbert space $ \mathcal H = L^2({\mathbb R}^n) \times L^2({\mathbb R}^n)$, where $g$, $V$ and $W$ are periodic time-dependent potentials, with period $T$. We denote by $(U(t,s))_{(t,s) \in {\mathbb R}\times {\mathbb R}}$ its associated propagator. By using a multiplier method, we rule out the existence of regular eigenvectors of the Floquet operator $U(T,0)$.

In this work we study the existence, uniqueness and asymptotic behavior in time of solutions for the dissipative Benjamin-Bona-Mahony equation in an exterior domain $\Omega \subset {{\mathbb R ^{n}}}$ ($n=2,3$). We obtain explicit polynomial decay rates for some norms of the solution which include the energy norm.

In this paper we prove a stability result for some classes of elliptic problems involving variable exponents. More precisely, we consider the Dirichlet problem for an elliptic equation in a domain $\Omega$, which is the $p$--Laplacian equation, $-\mbox{div}(|\nabla u|^{p-2} \nabla u) =f$, in a subdomain $\Omega_1$ of $\Omega$ and the Laplace equation, $-\Delta u = f$, in its complementary (that is, our equation involves the so-called $p(x)$--Laplacian with a discontinuous exponent). We assume that the right-hand side $f$ belongs to $L^\infty(\Omega)$. For this problem, we study the behaviour of the solutions as $p$ goes to $1$, showing that they converge to a function $u$, which is almost everywhere finite when the size of the datum $f$ is small enough. Moreover, we prove that this $u$ is a solution of a limit problem involving the $1$-Laplacian operator in $\Omega_1$. We also discuss uniqueness under a favorable geometry.

In this short note, we prove a refinement of bilinear local smoothing estimates of Airy solutions, when the frequency support of two wave are separated. As an application we prove a smoothing property of a bilinear form.

Given $\alpha > 0$ and $f\in L^2(0,1)$, consider the following singular Sturm-Liouville equation: \[ \left\lbrace\begin{aligned} -(x^{2\alpha}u'(x))'+u(x) & =f(x) \ \hbox{ a.e. on } (0,1),\\ u(1) & =0. \end{aligned}\right. \] We prove existence of solutions under (weighted) non-homogeneous boundary conditions at the origin.

In this paper we consider a free boundary problem (a modified Umeda's model) related to amoeba motion. We show the existence of time-global classical solutions and study their behavior in the spherically symmetric case. To this end, we carefully trace the trajectory of solutions, and find a positive invariant region to show the boundedness of solutions. We observe that the properties of the solutions agree with the real motion of amoeba.

We consider the Cauchy problem for the 2D and 3D Klein-Gordon Schrödinger system. In 2D we show local well posedness for Schrödinger data in $H^s$ and wave data in $H^{\sigma} \times H^{\sigma -1}$ for $s=-1/4 \, +$ and $\sigma = -1/2$, whereas ill posedness holds for $s < - 1/4$ or $\sigma < -1/2$, and global well-posedness for $s\ge 0$ and $s-\frac{1}{2} \le \sigma < s+ \frac{3}{2}$. In 3D we show global well posedness for $s \ge 0$, $ s - \frac{1}{2} < \sigma \le s+1$. Fundamental for our results are the studies by Bejenaru, Herr, Holmer and Tataru [2], and Bejenaru and Herr [3] for the Zakharov system, and also the global well-posedness results for the Zakharov and Klein-Gordon-Schrödinger system by Colliander, Holmer and Tzirakis [5].

We derive local boundedness estimates for weak solutions of a large class of second-order quasilinear equations. The structural assumptions imposed on an equation in the class allow vanishing of the quadratic form associated with its principal part and require no smoothness of its coefficients. The class includes second-order linear elliptic equations as studied in [5] and second-order subelliptic linear equations as in [8, 9]. Our results also extend ones obtained by J. Serrin [7] concerning local boundedness of weak solutions of quasilinear elliptic equations.