In the case of radial weights, the Mizohata–Takeuchi conjecture was independently verified by Barceló–Ruiz–Vega and Carbery–Soria. We discuss certain aspects of these influential papers and make connections with our own recent work on optimal constants for Kato-smoothing estimates.

We consider the asymptotic behavior of solutions for the Schrödinger equation with a Dirac delta potential and a gauge invariant power nonlinearity:\[(\delta{\text -}\mathrm{NLS}) \qquad \qquad \qquad i\partial_tu + \frac{1}{2} \partial_x^2 u - q\delta u = \lambda |u|^{p-1}u, \quad t,x \in \mathbb{R}$,\]where $u = u(t, x)$ is a $\mathbb{C}$-valued unknown function of $(t, x)$, $q \in \mathbb{R}\backslash\{0\}$ and $\lambda \in \mathbb{C} \backslash \{0\}$ are constants, and $\delta = \delta (x)$ is the Dirac measure centered at the origin $x = 0$. Our aim in the present paper is to prove an asymptotics of the solution to ($\delta$-NLS) and $L_x^2$-norm of the solution tends to 0 as $t \rightarrow \pm \infty$, in the repulsive, dissipative and long-range case:\[q \gt 0,\ \Im \lambda \lt 0 \mbox{ and } p = 3 \mbox{ or } p \lt 3 \mbox{ (close to 3)},\]for an arbitrarily prescribed final data in a lower order weighted Sobolev space.

Our proof of this result is based on a factorization technique for the linear solution ($\lambda = 0$) (Proposition 3.2) and properties of the distorted Fourier transformation (Proposition 4.1).

This paper is devoted to a proof of the conjecture in Takamura [16] on the lower bound of the lifespan of solutions to semilinear wave equations in two space dimensions. The result is divided into two cases according to the total integral of the initial speed.

We consider the Cauchy problem for systems of semilinear wave equations with small initial data in two and three space dimensions. We first present some recent results on the global existence and the asymptotic behavior of solutions under weaker conditions than the so-called null condition. Then, restricting our attention to two-component systems with nonlinearity of the critical power, we will discuss the asymptotic behavior in more details, and show that the asymptotic behavior observed in two space dimensions can be more complicated than that in three space dimensions.

We study a system of nonlinear Schrödinger equations with cubic interactions in one space dimension. The orbital stability and instability of semitrivial standing wave solutions are studied for both non-degenerate and degenerate cases.

We consider the thermo-elastic problem in the homogeneous isentropic material. After introducing the Helmholtz free energy of the thermo-elastic body and deriving a first approximation of the motion of elastic body with thermal effect, we show an $L^p$-type dissipative-dispersive estimate of Nishihara-type for the linearized equations and it shows that the solution is asymptotically decomposed into solutions to a linear heat equation, a solution to a linear wave equation of exponentially decaying and a diffusive wave ([6]). Then the sharp asymptotic behavior of the solutions to linearized thermo-elastic equations is shown for the coupled elastic-thermal system. As a by-product, we also obtain the Nishihara-type $L^p$ decay estimate for damped wave equation as the limiting case.

This manuscript presents some results on the decay estimate and asymptotic behavior of small solutions to the Cauchy problem of 1D Schrödinger equations with a sub-critical dissipative nonlinearity. Our aim is to determine the explicit lower bound of the nonlinear power for which certain a priori estimate of the solution works well.

The purpose of this paper is to shed light on the fact that the global solvability for the quadratically perturbed wave equation with small initial data in two space dimension can be shown by using only a restricted set of vector fields associated with the space-time translation and spatial rotations. As a by-product, we establish almost best possible decay estimates related to the above vector fields, as well as the tangential derivatives to the forward light cones.

We prove global existence of small solutions to the initial value problem for a class of cubic derivative nonlinear Schrödinger systems with the masses satisfying suitable non-resonance relations. The large-time asymptotics of the solutions are also shown. This work is intended to provide a counterpart of the previous paper [20] in which the mass resonance case was treated.

We shall study special properties of solutions to the IVP associated to the Camassa-Holm equation on the line related to the regularity and the decay of solutions. The first aim is to show how the regularity on the initial data is transferred to the corresponding solution in a class containing the “peakon solutions”. In particular, we shall show that the local regularity is similar to that exhibited by the solution of the inviscid Burger's equation with the same initial datum. The second goal is to prove that the decay results obtained in [17] extend to the class of solutions considered here.

We study the Hardy type inequalities in the framework of equalities. We present equalities which immediately imply Hardy type inequalities by dropping the remainder term. Simultaneously we give a characterization of the class of functions which makes the remainder term vanish. A point of our observation is to apply an orthogonality properties in general Hilbert space, and which gives a simple and direct understanding of the Hardy type inequalities as well as the nonexistence of nontrivial extremizers.

In this article, we consider mass-subcritical Hartree equation. Scattering problem is treated in the framework of weighted spaces. We first establish basic properties such as local-wellposedness and criteria for finite-time blowup and scattering. Then, the first result is that uniform in time bound in critical weighted norm implies scattering. The proof is based on the concentration compactness/rigidity argument initiated by Kenig and Merle. By using the argument, existence of a threshold solution between small scattering solutions and other solutions is also deduced for the focusing model, which is the second result. The threshold is neither ground state nor any other standing wave solutions, as is known for the power type NLS equation.

We consider the nonlinear Schrödinger equations in the de Sitter spacetime. We first show some derivation of the Einstein equation and several models of the universe for general dimensions and complex metrics. Then some models of the uniform and isotropic universe are considered based on the Einstein equation. After deriveing nonlinear Klein-Gordon equations in de Sitter spacetime, and the nonlinear Schrödinger equation in de Sitter spacetime is derived as their nonrelativistic limits. Furthermore we consider the Cauchy problem of the Schrödinger equations in Sobolev spaces. Some effects of spatial variation on the problem are remarked.

We review some recent results on the large time asymptotic behavior of solutions to the Cauchy problem for the fourth-order nonlinear Schrödinger equation \[\Bigg\{ \begin{gather} i\partial_{t}u + \mu\partial_{x}^{2}u + \alpha \partial_{x}^{4}u = N (u), \ (t,x) \in \mathbb{R}^{+} \times \mathbb{R}, \\ u ( 0, x) = u_{0} ( x), \ x \in \mathbb{R},\end{gather}\] with different types of the nonlinearity $N(u)$. We use the factorization technique originated from paper [30] for the free Schrödinger evolution group. This factorization formula appears very useful for studying the global existence and the large time asymptotic behavior of solutions to the nonlinear Schrödinger equations. Also it helps us to consider the problem in lower order Sobolev spaces. Later it was found that similar factorization technique can be developed for other dispersive equations, such as the nonlinear Klein-Gordon equation, fourth-order nonlinear Schrödinger equation, fractional order cubic nonlinear Schrödinger equation, reduced Ostrovsky equation, etc. By using the factorization formula, we divide the nonlinear term into the main and remainder terms, then we find the appropriate phase corrections in the large time asymptotic formula.

This paper is concerned with the scattering operator $S$ for the one dimensional Dirac equation with a quintic nonlinearity. It has been proved that $S$ can be defined on a neighborhood of 0 in the Sobolev space $H^\kappa (\mathbb{R};\mathbb{C}^2)$ for any $\kappa > 3/4$. In the present paper, we prove that for any $M \in \mathbb{N}$ and $s \ge \max\{ \kappa,M \}$, there exists some neighborhood $U$ of 0 in the weighted Sobolev space $H^{s,M}(\mathbb{R};\mathbb{C}^2)$ such that $S(U) \subset H^{s,M}(\mathbb{R};\mathbb{C}^2)$.

This paper is concerned with blowup for a complex Ginzburg–Landau equation $e^{-i\theta} u_t = \Delta u + |u|^{\alpha}u$, where $-\frac{\pi}{2} \lt \theta \lt \frac{\pi}{2}$, $\alpha \gt 0$. It is shown that if $\alpha \lt \frac{4}{N}$ and the initial value has negative energy, then the solution blows up in the norm $\| \nabla \cdot \|_{L^2(\mathbb{R}^N)}$. This result is almost a direct consequence from the result by Cazenave, Dickstein and Weissler (SIAM J. Math. Anal. 45 (2013), 244–266). The result gives a contrast to the results for the nonlinear Schrödinger equation. Moreover, it is established that the solution blows up in finite time even when the initial value does not have negative energy. In the proof it is shown that the well known potential well method for the nonlinear heat equation can be applied to the present case. From these results this paper clarifies that a complex Ginzburg–Landau equation has some features of parabolic equations.

We consider the magnetic Schrödinger equation $iD_0 u = D_k D^k u$ with electro-magnetic potential $A = (A_\mu)$ in $\boldsymbol{R}^{1+n}$, where $n \ge 3$ and $D_\mu = \partial_\mu + iA_\mu$. Under the assumption $A \in \cap_{j=0}^1 C^j(0,T;H^{1-j}_n)$ with $\sum_\alpha \| F_{\mu\nu};L^\infty (0,T; L^n(Q_\alpha))\|^2 \lt \infty$, we prove the Kato type smoothing estimate $$\sup_\alpha \| u; L^2(0,T; H^{1/2} (Q_\alpha)\| \lesssim \| u(0,\cdot)\|_2,$$ where $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$ and $\{Q_\alpha\}_{\alpha \in \boldsymbol{Z}^n}$ is the family of unit cubes.

In this paper, we consider the Cauchy problem of the wave equation with time-dependent damping. We prove that if the damping is sufficiently strong, then the solution has the second order asymptotic expansion the same as the corresponding heat equation.