In this paper, we consider a biharmonic equation under the Navier boundary condition and with a nearly critical exponent $(P_ε): Δ^2u=u^{9-ε}, u>0$ in $Ω$ and $u=Δu=0$ on $\partialΩ$, where $Ω$ is a smooth bounded domain in $\R^5$ and $ε>0$. We study the asymptotic behavior of solutions of $(P_ε)$ which are minimizing for the Sobolev qutient as $ε$ goes to zero. We show that such solutions concentrate around a point $x_0\inΩ$ as $ε\to 0$, moreover $x_0$ is a critical point of the Robin's function. Conversely, we show that for any nondegenerate critical point $x_0$ of the Robin's function, there exist solutions concentrating around $x_0$ as $ε$ goes to zero.
We consider the focusing $\dot H^{s_c}$-critical biharmonic Schrödinger equation, and prove a global wellposedness and scattering result for the radial data $u_0\in H^2(\mathbb R^N)$ satisfying $ M(u_0)^{\frac{2-s_c}{s_c}}E(u_0)<M(Q)^{\frac{2-s_c}{s_c}}E(Q) $ and $ \|u_{0}\|^{\frac{2-s_c}{s_c}}_{2}\|Δu_{0}\|_{2}<\|Q\|^{\frac{2-s_c}{s_c}}_{2}\|ΔQ\|_{2}, $ where $s_c\in(0,2)$ and $Q$ is the ground state of $Δ^2Q+(2-s_c)Q-|Q|^{p-1}Q=0$.
We study the focusing mass-subcritical biharmonic nonlinear Schrödinger equation (BNLS) on the product space $\mathbb{R}_x^d\times\mathbb{T}_y^n$. Following the crucial scaling arguments introduced in \cite{TTVproduct2014} we establish existence and stability results for the normalized ground states of BNLS. Moreover, in the case where lower order dispersion is absent, we prove the existence of a critical mass number $c_0\in(0,\infty)$ that sharply determines the $y$-dependence of the deduced ground states. In the mixed dispersion case, we encounter a major challenge as the BNLS is no longer s…
We consider the inhomogeneous biharmonic nonlinear Schrödinger equation (IBNLS) $$ i u_t +Δ^2 u+λ|x|^{-b}|u|^αu = 0, $$ where $λ=\pm 1$ and $α$, $b>0$. We show local and global well-posedness in $H^s(\mathbb{R}^N)$ in the $H^s$-subcritical case, with $s=0,2$. Moreover, we prove a stability result in $H^2(\mathbb{R}^N)$, in the mass-supercritical and energy-subcritical case. The fundamental tools to prove these results are the standard Strichartz estimates related to the linear problem.
We consider critical points $u:Ω\to N$ of the bi-energy \[ \int_Ω|Δu|^2\,d x, \] where $Ω\subset\mathbb{R}^m$ is a bounded smooth domain of dimension $m\ge 5$ and $N\subset\mathbb{R}^L$ a compact submanifold without boundary. More precisely, we consider variationally biharmonic maps $u\in W^{2,2}(Ω,N)$, which are defined as critical points of the bi-energy that satisfy a certain stationarity condition up to the boundary. For weakly convergent sequences of variationally biharmonic maps, we demonstrate that the only obstruction that can prevent the strong compactness up to the boundary is the pr…
In this paper we consider the following biharmonic equation with critical exponent $P_ε$ : $Δ^2 u= Ku^{(n+4)/(n-4)-ε}, u>0$ in $Ω$ and $u=Δu=0$ on $\partialΩ$, where $Ω$ is a domain in $R^n$, $n\geq 5$, $ε$ is a small positive parameter and $K$ is smooth positive function. We construct solutions of $P_ε$ which blow up and concentrate at strict local maximum of $K$ either at the boundary or in the interior of $Ω$. We also construct solutions of $P_ε$ concentrating at an interior strict local minimum of $K$. Finally, we prove a nonexistense result for the corresponding supercritical problem whic…
We consider the semilinear heat equation with a superlinear power nonlinearity in the Sobolev subcritical range. We construct a solution which blows up in finite time only at the origin, with a completely new blow-up profile, which is cross-shaped. Our method is general and extends to the construction of other solutions blowing up only at the origin, with a large variety of blow-up profiles, degenerate or not.
We introduce a straightforward method to analyze the blow-up of systems of ordinary differential inequalities, and apply it to study the blow- up of solutions to a weakly coupled system of semilinear heat equations. We prove that the solution blows up in a finite time under the subcritical condition. Moreover, we give estimates of lifespan of the solution and lower estimates of the blow-up rate for a localized average of each component.
Using dual method we establish the existence of nodal ground state solution for the following class of problems $$ \left\{ \begin{array}{l} Δ^2 u = f(u), \quad \mbox{in} \quad Ω, \\ u =Bu=0,\quad\mbox{on} \quad \partial Ω\end{array} \right. $$ where $Δ^2$ is the biharmonic operator, $B=Δ$ or $B=\dfrac{\partial}{\partial ν}$ and $f$ is a $C^1$-function having subcritical growth.
We study $O(d)$-equivariant biharmonic maps in the critical dimension. A major consequence of our study concerns the corresponding heat flow. More precisely, we prove that blowup occurs in the biharmonic map heat flow from $B^4(0, 1)$ into $S^4$. To our knowledge, this was the first example of blowup for the biharmonic map heat flow. Such results have been hard to prove, due to the inapplicability of the maximum principle in the biharmonic case. Furthermore, we classify the possible $O(4)$-equivariant biharmonic maps from $\mathbf{R}^4$ into $S^4$, and we show that there exists, in contrast to…
In this paper, we study the blow up of a sequence of (both extrinsic and intrinsic) biharmonic maps in dimension four with bounded energy and show that there is no neck in this process. Moreover, we apply the method to provide new proofs to the removable singularity theorem and energy identity theorem of biharmonic maps.
Using variational methods, we establish existence of multi-bump solutions for the following class of problems $$ \left\{ \begin{array}{l} Δ^2 u +(λV(x)+1)u = f(u), \quad \mbox{in} \quad \mathbb{R}^{N}, u \in H^{2}(\mathbb{R}^{N}), \end{array} \right. $$ where $N \geq 1$, $Δ^2$ is the biharmonic operator, $f$ is a continuous function with subcritical growth and $V : \mathbb{R}^N \rightarrow \mathbb{R}$ is a continuous function verifying some conditions.
We mainly consider the focusing biharmonic Schrödinger equation with a large radial repulsive potential $V(x)$: \begin{equation*} \left\{ \begin{aligned} iu_{t}+(Δ^2+V)u-|u|^{p-1}u=0,\;\;(t,x) \in {\bf{R}\times{\bf{R}}^{N}}, u(0, x)=u_{0}(x)\in H^{2}({\bf{R}}^{N}), \end{aligned}\right. \end{equation*} If $N>8$, \ $1+\frac{8}{N}<p<1+\frac{8}{N-4}$ (i.e. the $L^{2}$-supercritical and $\dot{H}^{2}$-subcritical case ), and $\langle x\rangle^β\big(|V(x)|+|\nabla V(x)|\big)\in L^\infty$ for some $β>N+4$, then we firstly prove a global well-posedness and scattering result for the radial data $u_0\in …
In this paper, we consider the well-posedness of the inhomogeneous nonlinear biharmonic Schrödinger equation with spatial inhomogeneity coefficient $K(x)$ behaves like $\left|x\right|^{-b}$ for $0<b<\min \left\{\frac{N}{2},4\right\} $. We show the local well-posedness in the whole $H^s$-subcritical case, with $0<s\le2$. The difficulties of this problem come from the singularity of $K(x)$ and the lack of differentiability of the nonlinear term. To resolve this, we derive the bilinear Strichartz's type estimates for the nonlinear biharmonic Schrödinger equations in Besov spaces.
We use asymptotic analysis and numerical simulations to study peak-type singular solutions of the supercritical biharmonic NLS. These solutions have a quartic-root blowup rate, and collapse with a quasi self-similar universal profile, which is a zero-Hamiltonian solution of a fourth-order nonlinear eigenvalue problem.
This paper is concerned with a biharmonic equation under the Navier boundary condition with nearly critical exponent. We study the asymptotic behavior os solutions which are minimizing for the Sobolev quatient. We show that such solutions concentrate around an interior point which is a critical point of the Robin's function. Conversely, we show that for any nondegenerate critical point fo the Robin's function, the exist solutions concentrating around such a point. Finally, we prove that, in contrast with what happened in the subcritical equation, the supercritical problem has no solutions whic…
We consider singular solutions of the biharmonic NLS. In the L^2-critical case, the blowup rate is bounded by a quartic-root power law, the solution approaches a self-similar profile, and a finite amount of L^2-norm, which is no less than the critical power, concentrates into the singularity ("strong collapse"). In the L^2-critical and supercritical cases, we use asymptotic analysis and numerical simulations to characterize singular solutions with a peak-type self-similar collapsing core. In the critical case, the blowup rate is slightly faster than a quartic-root, and the self-similar profile…
We consider the focusing energy subcritical nonlinear wave equation $\partial_{tt} u - Δu= |u|^{p-1} u$ in ${\mathbb R}^N$, $N\ge 1$. Given any compact set $ E \subset {\mathbb R}^N $, we construct finite energy solutions which blow up at $t=0$ exactly on $ E$. The construction is based on an appropriate ansatz. The initial ansatz is simply $U_0(t,x) = κ(t + A(x) )^{ -\frac {2} {p-1} }$, where $A\ge 0$ vanishes exactly on $ E$, which is a solution of the ODE $h'' = h^p$. We refine this first ansatz inductively using only ODE techniques and taking advantage of the fact that (for suitably chosen…
We prove a Brezis-Kato-type regularity result for weak solutions to the biharmonic nonlinear equation $$ Δ^2 u = g(x,u)\qquad\text{in }\mathbb{R}^N$$ with a Carathéodory function $g:\mathbb{R}^N\times \mathbb{R}\to \mathbb{R}$, $N\geq 5$. The regularity results give rise to the existence of ground state solutions provided that $g$ has a general subcritical growth at infinity. We also conceive a new biharmonic logarithmic Sobolev inequality $$ \int_{\mathbb{R}^N}|u|^2\log |u|\,dx\leq\frac{N}{8}\log \Big(C\int_{\mathbb{R}^N}|Δu|^2\,dx \Big), \quad\text{for } u \in H^2(\mathbb{R}^N), \; \int_{\ma…
In this work we study the nonnegative solutions of the elliptic system Δu=|x|^{a}v^δ, Δv=|x|^{b}u^μ in the superlinear case μδ>1, which blow up near the boundary of a domain of R^{N}, or at one isolated point. In the radial case we give the precise behavior of the large solutions near the boundary in any dimension N. We also show the existence of infinitely many solutions blowing up at 0. Furthermore, we show that there exists a global positive solution in R^{N}\{0}, large at 0, and we describe its behavior. We apply the results to the sign changing solutions of the biharmonic equation Δ^2 u=|…
This work concerns the semilinear wave equation in three space dimensions with a power-like nonlinearity which is greater than cubic, and not quintic (i.e. not energy-critical). We prove that a scale-invariant Sobolev norm of any non-scattering solution goes to infinity at the maximal time of existence. This gives a refinement on known results on energy-subcritical and energy-supercritical wave equation, with a unified proof. The proof relies on the channel of energy method, as in arXiv:1204.0031, in weighted scale-invariant Sobolev spaces which were introduced in arXiv:1506.00788. These space…
We study the possible blow-up behavior of solutions to the slightly subcritical elliptic problem with Hardy term \[ \left\{ \begin{aligned} -Δu-μ\frac{u}{|x|^2} &= |u|^{2^{\ast}-2-\varepsilon}u &&\quad \text{in } Ω, \\\ u &= 0&&\quad \text{on } \partialΩ, \end{aligned} \right. \] in a bounded domain $Ω\subset\mathbb{R}^N (N\ge7)$ with $0\inΩ$, as $μ,\varepsilon\to 0^+$. In \cite{BarGuo-ANS}, we obtained the existence of nodal solutions that blow up positively at the origin and negatively at a different point as $μ=O(ε^α)$ with $α>\frac{N-4}{N-2}$, $\varepsilon\to 0^+$. Here we prove the existe…
We present new singular solutions of the biharmonic nonlinear Schrodinger equation in dimension d and nonlinearity exponent 2σ+1. These solutions collapse with the quasi self-similar ring profile, with ring width L(t) that vanishes at singularity, and radius proportional to L^α, where α=(4-σ)/(σ(d-1)). The blowup rate of these solutions is 1/(3+α) for 4/d\leσ<4, and slightly faster than 1/4 for σ=4. These solutions are analogous to the ring-type solutions of the nonlinear Schrodinger equation.
We study the stochastic Nonlinear Schrödinger system with multiplicative white noise in energy space $H^1$. Based on deterministic and stochastic Strichartz estimates, we prove the local well-posedness and uniqueness of mild solution. Then we prove the global well-posedness in the mass subcritical case and the defocusing case. For the mass subcritical case, we also investigate the global existence when the $L^2$ norm of initial value is small enough. In addition, we study the blow-up phenomenon and give a sharp criteria.
The paper is concerned with the slightly subcritical elliptic problem with Hardy term \[ \left\{ \begin{aligned} -Δu-μ\frac{u}{|x|^2} &= |u|^{2^{\ast}-2-ε}u &&\quad \text{in } Ω, \\\ u &= 0&&\quad \text{on } \partialΩ, \end{aligned} \right. \] in a bounded domain $Ω\subset\mathbb{R}^N$ with $0\inΩ$, in dimensions $N\ge7$. We prove the existence of multi-bubble nodal solutions that blow up positively at the origin and negatively at a different point as $ε\to0$ and $μ=ε^α$ with $α>\frac{N-4}{N-2}$. In the case of $Ω$ being a ball centered at the origin we can obtain solutions with up to $5$ bubb…
We prove global well-posedness, scattering and blow-up results for energy-subcritical focusing nonlinear Schrödinger equations on the hyperbolic space. We show in particular the existence of a critical element for scattering for all energy-subcritical power nonlinearities. For mass-supercritical nonlinearity, we show a scattering vs blow-up dichotomy for radial solutions of the equation in low dimension, below natural mass and energy thresholds given by the ground states of the equation. The proofs are based on trapping by mass and energy, compactness and rigidity, and are similar to the ones …
Consider a family of smooth immersions $F(\cdot,t): M^n\to \mathbb{R}^{n+1}$ of closed hypersurfaces in $\mathbb{R}^{n+1}$ moving by the mean curvature flow $\frac{\partial F(p,t)}{\partial t} = -H(p,t)\cdot ν(p,t)$, for $t\in [0,T)$. We show that at the first singular time of the mean curvature flow, certain subcritical quantities concerning the second fundamental form, for example $\int_{0}^{t} \int_{M_{s}} \frac{\abs{A}^{n + 2}}{log (2 + \abs{A})} dμds,$ blow up. Our result is a log improvement of recent results of Le-Sesum, Xu-Ye-Zhao where the scaling invariant quantities were considered.
Consider the integral equation \begin{equation*} f^{q-1}(x)=\int_Ω\frac{f(y)}{|x-y|^{n-α}}dy,\ \ f(x)>0,\quad x\in \overline Ω, \end{equation*} where $Ω\subset \mathbb{R}^n$ is a smooth bounded domain. For $1<α<n$, the existence of energy maximizing positive solution in subcritical case $2<q<\frac{2n}{n+α}$, and nonexistence of energy maximizing positive solution in critical case $q=\frac{2n}{n+α}$ are proved in \cite{DZ2017}. For $α>n$, the existence of energy minimizing positive solution in subcritical case $0<q<\frac{2n}{n+α}$, and nonexistence of energy minimizing positive solution in crit…
We consider the slightly subcritical elliptic problem with Hardy term $$ \left\{ \begin{aligned} -Δu-μ\frac{u}{|x|^2} &= |u|^{2^{\ast}-2-ε}u &&\quad \text{in } Ω\subset\mathbb{R}^N, \\\ u &= 0&&\quad \text{on } \partial Ω, \end{aligned} \right. $$ where $0\inΩ$ and $Ω$ is invariant under the subgroup $SO(2)\times\{\pm E_{N-2}\}\subset O(N)$; here $E_n$ denots the $n\times n$ identity matrix. If $μ=μ_0ε^α$ with $μ_0>0$ fixed and $α>\frac{N-4}{N-2}$ the existence of nodal solutions that blow up, as $ε\to0^+$, positively at the origin and negatively at a different point in a general bounded domai…
We consider the semilinear heat equation $$\partial_t u -Δu =f(u), \quad (x,t)\in \mathbb{R}^N\times [0,T),\qquad (1)$$ with $f(u)=|u|^{p-1}u\log^a (2+u^2)$, where $p>1$ is Sobolev subcritical and $a\in \mathbb{R}$. We first show an upper bound for any blow-up solution of (1). Then, using this estimate and the logarithmic property, we prove that the exact blow-up rate of any singular solution of (1) is given by the ODE solution associated with (1), namely $u' =|u|^{p-1}u\log^a (2+u^2)$. In other terms, all blow-up solutions in the Sobolev subcritical range are Type I solutions. Up to our knowl…
In this paper, we study the following biharmonic equations:% $$ \left\{\aligned&Δ^2u-a_0Δu+(λb(x)+b_0)u=f(u)&\text{ in }\bbr^N,\\% &u\in\h,\endaligned\right.\eqno{(\mathcal{P}_λ)}% $$ where $N\geq3$, $a_0,b_0\in\bbr$ are two constants, $λ>0$ is a parameter, $b(x)\geq0$ is a potential well and $f(t)\in C(\bbr)$ is subcritical and superlinear or asymptotically linear at infinity. By the Gagliardo-Nirenberg inequality, we make some observations on the operator $Δ^2-a_0Δ+λb(x)+b_0$ in $\h$. Based on these observations, we give a new variational setting to $(\mathcal{P}_λ)$ for $a_0<0$. With this n…
In general, for higher order elliptic equations and boundary value problems like the biharmonic equation and the linear clamped plate boundary value problem neither a maximum principle nor a comparison principle or -- equivalently -- a positivity preserving property is available. The problem is rather involved since the clamped boundary conditions prevent the boundary value problem {from} being reasonably written as a system of second order boundary value problems. It is shown that, on the other hand, for bounded smooth domains $Ω\subset\mathbb{R}^n$, the negative part of the corresponding Gre…
We consider the semilinear wave equation \[ \partial_t^2 ψ-Δψ=|ψ|^{p-1}ψ\] for $1<p\leq 3$ with radial data in $\R^{3}$. This equation admits an explicit spatially homogeneous blow up solution $ψ^T$ given by $$ ψ^T(t,x)=κ_p (T-t)^{-\frac{2}{p-1}} $$ where $T>0$ and $κ_p$ is a $p$-dependent constant. We prove that the blow up described by $ψ^T$ is stable against small perturbations in the energy topology. This complements previous results by Merle and Zaag. The method of proof is quite robust and can be applied to other self-similar blow up problems as well, even in the energy supercritical cas…
We prove blow up results for the solution of the initial value problem with negative energy of the focusing mass-critical and supercritical nonlinear Schrödinger and the focusing energy-subcritical nonlinear wave equations with electromagnetic potential.
In this work we study the blow-up of solutions of a weakly coupled system of damped semilinear wave equations in the scattering case with power nonlinearities. We apply an iteration method to study both the subcritical case and the critical case. In the subcritical case our approach is based on lower bounds for the space averages of the components of local solutions. In the critical case we use the slicing method and a couple of auxiliary functions, recently introduced by Wakasa-Yordanov, to modify the definition of the functionals with the introduction of weight terms. In particular, we find …
We consider the following nonlinear Schrödinger equation with double power nonlinearities and an inverse power potential: \[ i\frac{\partial u}{\partial t}+Δu+|u|^{\frac{4}{N}}u+C_1|u|^{p-1}u+\frac{C_2}{|x|^{2σ}}u=0 \] in $\mathbb{R}^N$. From the classical argument, the solution with subcritical mass ($\left\|u_0\right\|_2<\left\|Q\right\|_2$) is global and bounded in $H^1(\mathbb{R}^N)$, where $Q$ is the ground state of the mass-critical problem. Previous results show the existence of a minimal-mass blow-up solution for the equation with $C_1>0$ and $C_2=0$ or $C_1=0$ and $C_2>0$ and investig…
This paper is concerned with finite blow-up solutions of the heat equation with nonlinear boundary conditions. It is known that a rate of blow-up solutions is the same as the self-similar rate for a Sobolev subcritical case. A goal of this paper is to construct a blow-up solution whose blow-up rate is different from the self-similar rate for a Sobolev supercritical case.
We consider the inhomogeneous biharmonic nonlinear Schrödinger (IBNLS) equation in $\mathbb{R}^N$, $$i \partial_t u +Δ^2 u -|x|^{-b} |u|^{2σ}u = 0,$$ where $σ>0$ and $b>0$. We first study the local well-posedness in $\dot H^{s_c}\cap \dot H^2 $, for $N\geq 5$ and $0<s_c<2$, where $s_c=\frac{N}{2}-\frac{4-b}{2σ}$. Next, we established a Gagliardo-Nirenberg type inequality in order to obtain sufficient conditions for global existence of solutions in $\dot H^{s_c}\cap \dot H^2$ with $0\leq s_c<2$. Finally, we study the phenomenon of $L^{σ_c}$-norm concentration for finite time blow up solutions w…
In this note we prove the instability by blow-up of the ground state solutions for a class of fourth order Schr\" odinger equations. This extends the first rigorous results on blowing-up solutions for the biharmonic NLS due to Boulenger and Lenzmann \cite{BoLe} and confirm numerical conjectures from \cite{BaFi, BaFiMa1, BaFiMa, FiIlPa}.
We show the local wellposedness of biharmonic wave maps with initial data of sufficiently high Sobolev regularity and a blow-up criterion in the sup-norm of the gradient of the solutions. In contrast to the wave maps equation we use a vanishing viscosity argument and an appropriate parabolic regularization in order to obtain the existence result. The geometric nature of the equation is exploited to prove convergence of approximate solutions, uniqueness of the limit, and continuous dependence on initial data.
Positive radial ground states for nonlinear biharmonic equations: maximum principles and expanding-domain approximation
Università degli Studi dell’Insubria（因苏布里亚大学）；RISM-Riemann International School of Mathematics（RISM黎曼国际数学学院）；China University of Geosciences（中国地质大学）；Università degli Studi di Udine（乌迪内大学）；Università di Pisa（比萨大学）
We consider the nonlinear Schrödinger equation with $L^{2}$-supercritical and $H^{1}$-subcritical power type nonlinearity. Duyckaerts and Roudenko and Campos, Farah, and Roudenko studied the global dynamics of the solutions with same mass and energy as that of the ground state. In these papers, finite variance is assumed to show the finite time blow-up. In the present paper, we remove the finite-variance assumption and prove a blow-up or grow-up result.
