We consider a strictly hyperbolic, genuinely nonlinear system of conservation laws in one space dimension. A sharp decay estimate is proved for the positive waves in an entropy weak solution. The result is stated in terms of a partial ordering among positive measures, using symmetric rearrangements and a comparison with a solution of Burgers' equation with impulsive sources.
In this paper, we prove the decay and scattering in the energy space for nonlinear Schrödinger equations with regular potentials in $\Bbb R^d$ namely, $i{\partial _t}u + Δu - V(x)u + λ|u|^{p - 1}u = 0$. We will prove decay estimate and scattering of the solution in the small data case when $1+\frac{2}{d}<p\le1+\frac{4}{d-2}$, $d\ge3$. The index $1+\frac{2}{d}$ is sharp for scattering concerning the result of W. Strauss [21].
In this paper, we consider the quadratic nonlinear Schrödinger system in three space dimensions. Our aim is to obtain sharp scattering criteria. Because of the mass-subcritical nature, it is difficult to do so in terms of conserved quantities. The corresponding single equation is studied by the second author and a sharp scattering criteria is established by introducing a distance from a trivial scattering solution, the zero solution. By the structure of the nonlinearity we are dealing with, the system admits a scattering solution which is a pair of zero solution and a linear solution. Taking t…
We study the nonlinear Schrödinger equation (NLS) with the quadratic nonlinearity $|u|^2$, posed on the two-dimensional torus $\mathbb{T}^2$. While the relevant $L^3$-Strichartz estimate is known only with a derivative loss, we prove local well-posedness of the quadratic NLS in $L^2(\mathbb{T}^2)$, thus resolving an open problem of thirty years since Bourgain (1993). In view of ill-posedness in negative Sobolev spaces, this result is sharp. We establish a crucial bilinear estimate by separately studying the non-resonant and nearly resonant cases. As a corollary, we obtain a tri-linear version …
This paper establishes sharp local regularity estimates for viscosity solutions of fully nonlinear parabolic free boundary problems with singular absorption terms. The main difficulties are due to the blow-up of the source along the free boundary and the lack of a variational structure. The proof combines the power of the Ishii-Lions method with intrinsically parabolic oscillation estimates. The results are new, even for second-order linear operators in nondivergence form.
This paper is concerned with the critical sharp traveling wave for doubly nonlinear diffusion equation with time delay, where the doubly nonlinear degenerate diffusion is defined by $\Big(\big|(u^m)_x\big|^{p-2}(u^m)_x\Big)_x$ with $m>0$ and $p>1$. The doubly nonlinear diffusion equation is proved to admit a unique sharp type traveling wave for the degenerate case $m(p-1)>1$, the so-called slow-diffusion case. This sharp traveling wave associated with the minimal wave speed $c^*(m,p,r)$ is monotonically increasing, where the minimal wave speed satisfies $c^*(m,p,r)<c^*(m,p,0)$ for any time del…
We study the asymptotic dynamics for solutions to a system of nonlinear Schrödinger equations with cubic interactions, arising in nonlinear optics. We provide sharp threshold criteria leading to global well-posedness and scattering of solutions, as well as formation of singularities in finite time for (anisotropic) symmetric initial data. The free asymptotic results are proved by means of Morawetz and interaction Morawetz estimates. The blow-up results are shown by combining variational analysis and an ODE argument, which overcomes the unavailability of the convexity argument based on virial-t…
Blow-up and sharp lifespan estimates to the weakly coupled system of structurally damped wave equations with critical nonlinearities
School of Mathematics and Computer Sciences, Hanoi National University of Education（河内国家教育大学数学与计算机科学学院）；Faculty of Mathematics and Informatics, Hanoi University of Science and Technology（河内科技大学数学与信息学院）；Department of Mathematics, Hanoi National University of Education（河内国家教育大学数学系）
We study the high energy estimate for the resolvent $R(λ)$ of the Laplacian on non-trapping asymptotically hyperbolic manifolds (AHM). In the literature, polynomial bound of the form $\|R(λ)\| = O(|λ|^{N})$ for $|λ|$ large and $λ\in \mathbb{C}$ in strips where $R(λ)$ is holomorphic was established for some $N > -1$. We prove the optimal bound $O(|λ|^{-1})$ under the non-positive sectional curvature assumption by taking into account the oscillatory behavior of the Schwartz kernel of the resolvent.
In this paper we study optimal lower and upper bounds for functionals involving the first Dirichlet eigenvalue $λ_{F}(p,Ω)$ of the anisotropic $p$-Laplacian, $1<p<+\infty$. Our aim is to enhance how, by means of the $\mathcal P$-function method, it is possible to get several sharp estimates for $λ_{F}(p,Ω)$ in terms of several geometric quantities associated to the domain. The $\mathcal P$-function method is based on a maximum principle for a suitable function involving the eigenfunction and its gradient.
We study the stability/instability of standing waves for the one dimensional nonlinear Schrödinger equation with double power nonlinearities: \begin{align*} &i\partial_t u +\partial_x^2 u -|u|^{p-1}u +|u|^{q-1}u=0, \quad (t,x)\in \mathbb{R}\times\mathbb{R} ,~1<p<q. \end{align*} When $q<5$, the stability properties of standing waves $e^{iωt}ϕ_ω$ may change for the frequency $ω$. A sufficient condition for yielding instability for small frequencies are obtained in previous results, but it has not been known what the sharp condition is. In this paper we completely calculate the explicit formula o…
We obtain optimal boundary and global regularity estimates for viscosity solutions of fully nonlinear elliptic equations whose ellipticity degenerates at the critical points of a given solution. We show that any solution is $C^{1,α}$ on the boundary of the domain, for an optimal and explicit $α$ given only in terms of the regularity of the boundary datum and the elliptic degeneracy degree, no matter how possibly low is the interior regularity for that class of equations. We also obtain sharp global estimates. Our findings are new even for model equations, involving only a degenerate Laplacian;…
We consider the problem of establishing nonlinear smoothing as a general feature of nonlinear dispersive equations, i.e. the improved regularity of the integral term in Duhamel's formula, with respect to the initial data and the corresponding regularity of the linear evolution, and how this property relates to local well-posedness. In a first step, we show how the problem generally reduces to the derivation of specific frequency-restricted estimates, which are multiplier estimates in the spatial frequency alone. Then, using a precise methodology, we prove these estimates for the specific cases…
This paper is concerned with the Cauchy problem of the quadratic nonlinear Schrödinger equation in $\mathbb{R} \times \mathbb{R}^2$ with the nonlinearity $η|u|^2$ where $η\in \mathbb{C} \setminus \{0\}$ and low regularity initial data. If $s < -1/4$, the ill-posedness result in the Sobolev space $H^{s}(\mathbb{R}^2)$ is known. We will prove the well-posedness in $H^s(\mathbb{R}^2)$ for $-1/2 < s < -1/4$ by assuming some angular regularity on initial data. The key tools are the modified Fourier restriction norm and the convolution estimate on thickened hypersurfaces.
We study the Lane-Emden system $$\begin{cases} -Δu=v^p,\quad u>0,\quad\text{in}~Ω, -Δv=u^q,\quad v>0,\quad\text{in}~Ω, u=v=0,\quad\text{on}~\partialΩ, \end{cases}$$ where $Ω\subset\mathbb{R}^2$ is a smooth bounded domain. In a recent work, we studied the concentration phenomena of positive solutions as $p,q\to+\infty$ and $|q-p|\leq Λ$. In this paper, we obtain sharp estimates of such multi-bubble solutions, including sharp convergence rates of local maxima and scaling parameters, and accurate approximations of solutions. As an application of these sharp estimates, we show that when $Ω$ is con…
We prove sharp regularity estimates for viscosity solutions of fully nonlinear parabolic equations of the form \begin{equation}\label{Meq}\tag{Eq} u_t- F(D^2u, Du, X, t) = f(X,t) \quad \mbox{in} \quad Q_1, \end{equation} where $F$ is elliptic with respect to the Hessian argument and $f \in L^{p,q}(Q_1)$. The quantity $κ(n, p, q):=\frac{n}{p}+\frac{2}{q}$ determines to which regularity regime a solution of \eqref{Meq} belongs. We prove that when $1< κ(n,p,q) < 2-ε_F$, solutions are parabolic-Hölder continuous for a sharp, quantitative exponent $0< α(n,p,q) < 1$. Precisely at the critical border…
We consider the initial value problem (IVP) associated to the cubic nonlinear Schrödinger equation with third-order dispersion \begin{equation*} \partial_{t}u+iα\partial^{2}_{x}u- \partial^{3}_{x}u+iβ|u|^{2}u = 0, \quad x,t \in \mathbb{R}, \end{equation*} for given data in the Sobolev space $H^s(\mathbb{R})$. This IVP is known to be locally well-posed for given data with Sobolev regularity $s>-\frac14$ and globally well-posed for $s\geq 0$ [3]. For given data in $H^s(\mathbb{R})$, $0>s> -\frac14$ no global well-posedness result is known. In this work, we derive an almost conserved quantity for…
We study decay of small solutions of the Born-Infeld equation in 1+1 dimensions, a quasilinear scalar field equation modeling nonlinear electromagnetism, as well as branes in String theory and minimal surfaces in Minkowski space-times. From the work of Whitham, it is well-known that there is no decay because of arbitrary solutions traveling to the speed of light just as linear wave equation. However, even if there is no global decay in 1+1 dimensions, we are able to show that all globally small $H^{s+1}\times H^s$, $s>\frac12$ solutions do decay to the zero background state in space, inside a …
This is the continuation of our previous work [5], where we introduced and studied some nonlinear integral equations on bounded domains that are related to the sharp Hardy-Littlewood-Sobolev inequality. In this paper, we introduce some nonlinear integral equations on bounded domains that are related to the sharp reversed Hardy-Littlewood-Sobolev inequality. These are integral equations with nonlinear term involving negative exponents. Existence results as well as nonexistence results are obtained.
We consider the 3D smectic energy $$\mathcal{E}_{ε}\left( u\right) =\frac{1}{2}\int_{Ω}\frac{1}{\varepsilon } \left( u_z-\frac{( u_x)^{2}+( u_y)^{2}}{2}\right) ^{2}+\varepsilon \left( u_{xx}+u_{yy}\right)^{2}\,dx\,dy\,dz. $$ The model contains as a special case the well-known 2D Aviles-Giga model. We prove a sharp lower bound on $\mathcal{E}_{\varepsilon}$ as $\varepsilon \to 0$ by introducing 3D analogues of the Jin-Kohn entropies. The sharp bound corresponds to an equipartition of energy between the bending and compression strains and was previously demonstrated in the physics literature onl…
We classify local minimizers of $\intσ_2+\oint H_2$ among all conformally flat metrics in the Euclidean $(n+1)$-ball, $4\leq n\leq 5$, for which the boundary has unit volume, subject to an ellipticity assumption. We also classify local minimizers of the analogous functional in the critical dimension $n+1=4$. If minimizers exist, this implies a fully nonlinear sharp Sobolev trace inequality. Our proof is an adaptation of the Frank--Lieb proof of the sharp Sobolev inequality, and in particular does not rely on symmetrization or Obata-type arguments.
Consider the heat equation with a nonlinear boundary condition $$ \partial_t u=Δu,\quad x\in{\bf R}^N_+,\,\,\,t>0,\qquad \partial_νu=u^p, \quad x\in\partial{\bf R}^N_+,\,\,\,t>0,\qquad u(x,0)=κψ(x),\quad x\in D:=\overline{{\bf R}^N_+}, $$ where $N\ge 1$, $p>1$, $κ>0$ and $ψ$ is a nonnegative measurable function in ${\bf R}^N_+ :=\{y\in{\bf R}^N:y_N>0 \}$. Let us denote by $T(κψ)$ the life span of solutions to this problem. We investigate the relationship between the singularity of $ψ$ at the origin and $T(κψ)$ for sufficiently large $κ>0$ and the relationship between the behavior of $ψ$ at the…
Existence of positive solution for a nonlinear elliptic equation with saddle-like potential and nonlinearity with exponential critical growth in $\mathbb{R}^{2}$
In this paper, we study a class of Schrödinger-Poisson (SP) systems with general nonlinearity where the nonlinearity does not require Ambrosetti-Rabinowitz and Nehari monotonic conditions. We establish new estimates and explore the associated energy functional which is coercive and bounded below on Sobolev space. Together with Ekeland variational principle, we prove the existence of ground state solutions. Furthermore, when the `charge' function is greater than a fixed positive number, the (SP) system possesses only zero solutions. In particular, when `charge' function is radially symmetric, w…
We study the stability of traveling waves of nonlinear Schrödinger equation with nonzero condition at infinity obtained via a constrained variational approach. Two important physical models are Gross-Pitaevskii (GP) equation and cubic-quintic equation. First, under a non-degeneracy condition we prove a sharp instability criterion for 3D traveling waves of (GP), which had been conjectured in the physical literature. This result is also extended for general nonlinearity and higher dimensions, including 4D (GP) and 3D cubic-quintic equations. Second, for cubic-quintic type sub-critical or critica…
